A Model for the Origin of
Homochirality
in Polymerization
and β Sheet Formation
Nathaniel Wagner, βoris Rubinov, Gonen Ashkenasy
Chirality
Chiral Molecule
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.
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
Chiral Amplification / Symmetry Breaking
Models and Experiments
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.”
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.
Kondepudi (1990)
Asakura (1995)
Ghadiri (2001)
Luisi (2001)
Buhse (2003)
Blackmond (2004)
Plasson (2004, 2007)
Viedma (2005)
Ribo (2008)
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:
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•
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•
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•
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 …
Toy Models
Polymerization leading to Homochirality
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
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)
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)
Self Replicating β-Sheet Forming Peptides
Simple peptides can form β sheets that serve as catalysts for self-replication.
E
N
Tn
E●N●Tn
Tn+1
Suggested Mechanism:
B. Rubinov, N. Wagner, H. Rapaport, G. Ashkenasy, Agnew. Chem. Int. Ed. 48, 6683 (2009)
Model and Simulation
(1) Irreversible Polymerization
(2) Reversible β Sheet Formation
(3) No Epimerization (Chirality Switching)
“Scoring” for β Sheets
• Stability of β sheets
proportional to no.
of hydrogen bonds
• Monomers prefer to
join peptides that
are part of β sheets
“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
Allow us to take the various scoring factors into account
1
Weighting Factor =
1 + e- w * (s-m)
k=0.3
k=10
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
Strength
1
Strength
Strength
k=0
1
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
-10
-8
-6
-4
-2
0
Score
2
4
6
8
10
0
-10
-8
-6
-4
-2
0
Score
2
4
6
8
10
0
-10
-8
-6
-4
-2
0
Score
2
4
6
8
10
s-m
s-m
s-m
No Weighting
(w=0)
Medium Weighting
High Weighting
(w)
Simulation Flow Chart
Concurrent processes (until monomers run out):
Polymerization
β Sheet Formation
β Sheet Breakup
Chiral Polymerization
Binomial distribution
D4 D3L D2L2 DL3 L4
Chiral Polymerization
Polymerization w = 3 β Sheet w = 3
Polymerization w = 0.3 β Sheet w = 3
Chiral Polymerization
Experimental Results (Lahav 2007)
Simulation Results
Non-Racemic Initial Conditions (20% ee)
20% ee
-20% ee
enantiomeric excess:
ee = ([L]-[D])/([L]+[D])
diastereomeric excess:
  j  k   L
de i 
j k i

j k i
j
D k 
 L j Dk 


Summary and Conclusions
• 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
N. Wagner, B. Rubinov, G. Ashkenasy, Chem. Phys. Chem. 12, 2771 (2011)
Special Thanks to …
Prof. Gonen Ashkenasy
Boris Rubinov
Prof. Meir Lahav
Maya Kleiman
Prof. Addy Pross ● Prof. Emmanuel Tannenbaum
Dr. Rivka Cohen-Luria ● Dr. Dima Lukatsky
Zehavit Dadon ● Rita Eisenberg ● Yosi Shamay ● Inbal Shumacher ● Lina Shtelman
Samaa Alesebi ● Dennis Ivnitski ● Valery Bourbo ● Nadia Levin ● Vered Zavaro
Yoav Raz ● Eran Itan
Prof. Avshalom Elitzur
Machon IYAR – Israel Institute for Advanced Study
A Model for the Origin of Homochirality in Polymerization and b 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:
m i
m
i  1
1
m 0i
m
i  1
 2 
m 0  i 1 m
i  1
m 0i
m
i  1
 1
Combining terms yields the difference equation:
m i
This can be expanded further:
m i

m 0  i 1
m 0i
 1  m
i  1

m 0  i 1
m 0i

1

 1  m

m 0i 
m 0 i 1
m 0  i 1
m 0i 
m 0  i 1

1


1

 1  m


m 0i 
m 0  i 1
m 0i2
i  2
i  3






The sum goes all the way to m0, yielding:
m i
m 0  i 1
m 0i

m 0  i 1
m 0  i 1

m 0  i 1
m 0i2


m 0  i 1
m 0 1
 1  m 
0
or equivalently:
m i
m 0  i 1
m 0i

m 0  i 1
m 0  i 1

m 0  i 1
m 0i2


m 0  i 1
m 0 1
m
m 0  i 1
0
m 0 1
This can be listed as a finite summation:
 m 0
m i   m 0  i  1 

 m 0 1
m o 1

jm oi
1

j
For very large m0 the summation approaches the integral:
 m 0
m i   m 0  i  1 

 m 0 1

m o 1


d j

j

1
m oi
and this can then be approximated by:

m 0 
m i   m 0  i  1   1  ln

m
0  i


We are interested in the step where the monomers run out, which is given by:
1  ln
m
0
m 0i
0
m
0
m 0i
e
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)
enantiomeric excess:
ee = ([L]-[D])/([L]+[D])
diastereomeric excess:
  j  k   L
de i 
j
j k i

j k i
D k 
 L j Dk 


isotacticity index:

iii 
j k i
j  k  L j D k 

j k i
20% ee
-20% ee
 L j Dk 


Open System Simulation
ΔLΔ , D
intake
Li , Di
r Li r ,Di
outtake

ΔliΔ , di
li , di
proportionally
increasing intake
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.
20% ee followed by 100 racemic ‘waves’ (1% intake)
  j  k   L
de i 
j k i

j k i
j
D k 
 L j Dk 


ee = ([L]-[D])/([L]+[D])
Peptide length = 3
Peptide length = 10
Time Evolution of Open Systems
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’