A Model for the Origin of Homochirality

in Polymerization and β Sheet Formation

Nathaniel Wagner, βoris Rubinov, Gonen Ashkenasy

Chirality

A molecule that has a nonidentical mirror image, which does not contain a plane of symmetry, is said to be chiral. The pair of molecules are called enantiomers.

Chiral Molecule

Naming Conventions

R (Rectus) vs. S (Sinister)

D (Dextro) vs. L (Levo)

Homochirality in Nature

• Active amino acids are all L

• Biologically relevant sugars are mostly D

• RNA and DNA are also homochiral

What is the origin of homochirality?

How is this related to the origin of life?

What came first – life or homochirality?

Suggested Origins of Homochirality

• Random initial fluctuations followed by chiral amplification• Evolution: racemic molecules too inefficient for biochemical life

processes• Extra-terrestial: L-amino acids came from outer space• Cosmic radiation: circularly polarized light delivered to primitive Earth by

comets or meteorites• Parity violation in weak interaction leads to a small energy shift between

enantiomers, resulting in a phase transition• Thermodynamic-kinetic feedback near equilibrium led to chiral separation• Enantioselective adsorption or reactions on chiral surfaces (e.g., quartz)• Magnetic fields or Vortex motion• Spontaneous mirror symmetry breaking as a cooperative phenomenon in

non-linear dissipative systems

Frank Model (1953)Proposed a reaction scheme where chiral autocatalysis – in which each enantiomer catalyzes its own formation and suppresses the production of the opposite enantiomer - amplifies small statistical fluctuations, leading to large enantiomeric excess. Concluded: “A laboratory demonstration may not be impossible.”

Kondepudi (1990)Asakura (1995)Ghadiri (2001)Luisi (2001)Buhse (2003)Blackmond (2004)Plasson (2004, 2007)Viedma (2005)Ribo (2008)

Chiral Amplification / Symmetry BreakingModels and Experiments

Soai Reaction (1990, 1995) Designed and implemented an asymmetric autocatalytic reaction system, where a small enantiomeric excess yielded a much larger excess at the end of the reaction.

Lahav et al (2007-2009)

Studied polymerization of amino acids in aquaeous solutions, and observed long homochiral chains when antiparallel β sheets were formed.

Theoretical Explanation of Lahav et al

Goals:

•Gain insight into kinetic model

•Simulate the potential role of β sheets in peptide length distributions

•Highlight the parameters that affect homochiral polymerization

•Probe the interplay between the various competing processes

•Investigate open systems

•Provide “recipes” for further experiments

•Further speculation …

A toy model is a simplified set of objects and equations used to understand complex mechanisms. This is usually done by reducing the number of dimensions or variables, while leaving in the essential details, in order to reproduce the main qualitative effects.

Sandars (2003)

Wattis and Coveney (2005)

Brandenburg et al (2005, 2007)

Kafri, Markovitch, Lancet (2010) Chiral selection resulting from the GARD model

Blanco and Hochberg (2011) Explained symmetry breaking in Lahav’s chiral crystallization experimental results

Toy ModelsPolymerization leading to Homochirality

Brack (1979, 2007) Ulijn (2005)

Maury (2009)

Otto (2010)

β Sheets

It has been suggested that β sheet peptide structures have played a critical role in self-replication, homochirality, and the origin of life.

Brack (1979, 2007) Ulijn (2005)

Maury (2009)

Otto (2010)

Lahav et al (2007-2009)

β Sheets

It has been suggested that β sheet peptide structures have played a critical role in self-replication, homochirality, and the origin of life.

Self Replicating β-Sheet Forming Peptides

B. Rubinov, N. Wagner, H. Rapaport, G. Ashkenasy, Agnew. Chem. Int. Ed. 48, 6683 (2009)

Suggested Mechanism:

TnE●N●Tn Tn+1

E N

Simple peptides can form β sheets that serve as catalysts for self-replication.

Model and Simulation

(2) Reversible β Sheet Formation

(1) Irreversible Polymerization

(3) No Epimerization (Chirality Switching)

“Scoring” for β Sheets

• Monomers prefer to join peptides that are part of β sheets

• Stability of β sheets proportional to no. of hydrogen bonds

“Scoring” for Chirality

• Polymerization favors adjacent units of identical chirality

• β sheets strengthened by having units of opposite chirality in adjacent strands

• Monomers prefer to join β sheets where chirality of adjacent strands is opposite

Sigmoid Weighting Functions

1

1 + e- w * (s-m)

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Score

Str

engt

h

k=0.3

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Score

Str

engt

h

k=10

No Weighting( w = 0 )

High Weighting( w )

Medium Weighting

s-m s-ms-m

Weighting Factor =

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Score

Str

engt

h

k=0

Allow us to take the various scoring factors into account

Simulation Flow Chart

Concurrent processes (until monomers run out):

Polymerization β Sheet Formation β Sheet Breakup

Chiral Polymerization

D4 D3L D2L2 DL3 L4

Binomial distribution

Chiral Polymerization

Polymerization w = 3 β Sheet w = 3 Polymerization w = 0.3 β Sheet w = 3

Chiral Polymerization

Simulation ResultsExperimental Results (Lahav 2007)

Non-Racemic Initial Conditions (20% ee)

j kj k i

i

j kj k i

j k L D

deL D

ee = ([L]-[D])/([L]+[D])

20% ee-20% ee

enantiomeric excess: diastereomeric excess:

• Model:• Irreversible polymerization, reversible β sheet formation• No epimerization (chirality switching)

• Results:• Non-random distribution of peptide lengths• Tendency to form long isotactic peptide diastereoisomers• Consistent with Lahav et al’s experimental results• Critically depend on formation of β sheets• Hold for initially racemic or nonracemic systems• Hold for open or closed systems

• Possible Implications:• Scenario where Origin of Chirality preceded Origin of Life• Explanation for increase in chirality with biochemical complexity• Potential description of origin of chiroselectivity in biological systems

Summary and Conclusions

N. Wagner, B. Rubinov, G. Ashkenasy, Chem. Phys. Chem. 12, 2771 (2011)

Special Thanks to…

Prof. Gonen Ashkenasy Prof. Meir Lahav Boris Rubinov

Maya Kleiman

Prof. Addy Pross ● Prof. Emmanuel TannenbaumDr. Rivka Cohen-Luria ● Dr. Dima LukatskyZehavit Dadon ● Rita Eisenberg ● Yosi Shamay ● Inbal Shumacher ● Lina ShtelmanSamaa Alesebi ● Dennis Ivnitski ● Valery Bourbo ● Nadia Levin ● Vered Zavaro Yoav Raz ● Eran Itan

Prof. Avshalom ElitzurMachon IYAR – Israel Institute for Advanced Study

A Model for the Origin of Homochirality in Polymerization and Sheet Formation

Nathaniel Wagner, Boris Rubinov and Gonen Ashkenasy

Abstract

The origin of homochirality in living systems is an open question closely related to the origin of life. Several explanations and models have been proposed, while various experimental systems demonstrating increasing chirality have been discovered using autocatalysis, crystallization or polymerization. In one particular approach, Lahav et al1 studied the chiral amplification obtained during peptide formation by polymerization and β sheet formation of amino acid building blocks. Consequently, we have introduced a simple model and stochastic simulation for this system2, showing the crucial effects of the β sheets on the distributions of peptide lengths. When chiral affinities are included, racemic β sheets of alternating homochiral strands lead to the formation of chiral peptides whose isotacticity increases with length, consistent with the experimental results. The tendency to form isotactic peptides is shown for both initially racemic and initially nonracemic systems, as well as for closed and open systems. We suggest that these or similar mechanisms may explain the origin of chiroselectivity in prebiotic environments.

1. I. Rubinstein, R. Eliash, G. Bolbach, I. Weissbuch and M. Lahav, Angew. Chem. Int. Ed. 46, 3710 (2007); I. Weissbuch, R. A. Illos, G. Bolbach and M. Lahav, Accounts Chem. Res. 42, 1128 (2009).

2. N. Wagner, B. Rubinov and G. Ashkenasy, Chem. Phys. Chem. 12, 2771 (2011).

Achiral Polymerization

a) w = 0 m = 0 b) w = 3 m = 2

c) w = 2 m = 5 d) w = 5 m = 3

Computation of Average Length for Random Achiral Polymerization

The average polymer length for random achiral polymerization, when all the monomers have been polymerized, can be computed as follows.

Let m0 be the initial number of monomers, and mi the number left after step i, and let ni be the total number of

independent units (monomers and polymers) available after step i. Since we begin only with monomers, n0 = m0.

The average polymer length after any step i will be m0 / ni. After each step one less unit is available, so ni

decreases by 1, yielding ni = m0 – i. On the other hand, mi decreases either by 2 if a monomer joins another

monomer, or by 1 if a monomer joins a polymer. At each step i, a monomer may join another monomer with probability (mi-1 -1) / (ni-1 -1), or a polymer with probability (ni-1 - mi-1) / (ni-1 -1). On average, the number of

monomers then becomes:

1 0 1

1 1

0 0

1 12 1

i i

i i i

m m i mm m m

m i m i

Combining terms yields the difference equation:

0

1

0

11i i

m im m

m i

This can be expanded further:

0 0

2

0 0

11 1

1i i

m i m im m

m i m i

0 0 0

3

0 0 0

1 11 1 1

1 2i

m i m i m im

m i m i m i

The sum goes all the way to m0, yielding:

0 0 0 0

0

0 0 0 0

1 1 1 11

1 2 1i

m i m i m i m im m

m i m i m i m

or equivalently:

0 0 0 0 0

0

0 0 0 0 0

1 1 1 1 1

1 2 1 1i

m i m i m i m i m im m

m i m i m i m m

This can be listed as a finite summation:

o

0

0

0 o

1 11

1i

m

j m i

mm m i

m j

For very large m0 the summation approaches the integral:

o

0

0

0o

11

11

i

m

m i

mm m i d j

m j

and this can then be approximated by:

0

0

0

1 1 lni

mm m i

m i

We are interested in the step where the monomers run out, which is given by:

0 0

0 0

1 ln 0m m

em i m i

The last term is just the expression for the average length, yielding an average polymer length of e, independent of m0.

Non-Racemic Initial Conditions (20% ee)

j kj k i

i

j kj k i

j k L D

deL D

j kj k i

i

j kj k i

j k L D

iiL D

ee = ([L]-[D])/([L]+[D])

20% ee-20% ee

enantiomeric excess:

diastereomeric excess:

isotacticity index:

Open System Simulation

Li , Di

r Li , r Di li , di

ΔL , ΔD Δli , Δdi

Schematic diagram illustrating an open continuous reaction system, and its equivalent model as implemented in the simulation, calculated in order to keep the correct proportions.

intake outtake proportionally increasing intake

20% ee followed by 100 racemic ‘waves’ (1% intake)

j kj k i

i

j kj k i

j k L D

deL D

ee = ([L]-[D])/([L]+[D])

Peptide length = 3

Peptide length = 10

a) ee = 0 followed by 100 racemic ‘waves’b) ee = 20% followed by 100 racemic ‘waves’c) ee = 50% followed by 100 racemic ‘waves’

Time Evolution of Open Systems

Peptide length = 3

Peptide length = 10