∎

11institutetext:  1Nordita, KTH Royal Institute of Technology and Stockholm
University, Stockholm, Sweden
2Department of Astronomy, AlbaNova University Center, Stockholm, Sweden
3McWilliams Center for Cosmology, Carnegie Mellon University, Pittsburgh, PA
15213, USA
Correspondence: Nordita, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden
11email<EMAIL_ADDRESS>Tel: +46 8 5537 8707, mobile: +46 73 270 4376
http://orcid.org/0000-0002-7304-021X

# Homochirality: a prerequisite or consequence of life?

Axel Brandenburg

###### Abstract

Many of the building blocks of life such as amino acids and nucleotides are
chiral, i.e., different from their mirror image. Contemporary life selects and
synthesizes only one of two possible handednesses. Chiral molecules also tend
to be optically active and rotate polarized light in a left-handed sense for
many proteins and in a right-handed sense for many nucleotides and sugars. In
an abiotic environment, however, there are usually equally many left- and
right-handed molecules, so we talk about a racemic mixture. If homochirality
was a prerequisite of life, there must have been physical or chemical
circumstances that led to the selection of a certain preference. Conversely,
if it was a consequence of life, we must identify possible pathways for
accomplishing a transition from a racemic to a homochiral chemistry. There has
been significant progress on both approaches. One of the four elementary
forces of nature, the weak force, responsible for the decay of free neutrons,
for example, does give rise to a preference for chemical reactions between
molecules of a certain chirality, but the effect is very small compared to
that of random fluctuations that are always present. On the other hand, amino
acids from certain meteorites suggest a preference for the left-handed amino
acids – at least for some of them – although the question of contamination is
not fully ruled out. This could be explained by polarized light, which can
give rise to a selection of a net handedness of biomolecules, but the effect
is again small. Depending on the mechanism that is responsible for generating
polarized light, either both signs or only one sign of polarization are
possible. After a detailed discussion of these ideas and the observational
evidence, we also review alternative ideas where homochirality of any
handedness could emerge as a consequence of the first polymerization events of
nucleotides in an emerging RNA world. In those proposals, autocatalysis was
thought to be a crucial ingredient, but recent studies show that this is not
necessarily the case. Also, the effect of enantiomeric cross inhibition, the
termination of polymerization with monomers of the opposite chirality, may not
be detrimental, as was originally thought, but it may instead be beneficial.
There are indeed mechanisms that can produce full homochirality through the
combination of both autocatalysis and enantiomeric cross inhibition. These
mechanisms are not limited to nucleotides, but can also occur for peptides, as
a precursor to the RNA world. The question of homochirality is, in this sense,
intimately tied to the origin of life. Future Mars missions may be able to
detect biomolecules of extant or extinct life. We will therefore also discuss
possible experimental setups for determining the chirality of primitive life
forms in situ on Mars.

###### Keywords:

DNA polymerization, enantiomeric cross-inhibition, origin of homochirality.
Revision: 1.79

##  3.1 Introduction

The occurrence of handedness in biology is not uncommon. The difference
between our left and right hands is the most obvious occurrence in the
macroscopic world. In ancient Greek, the word $\chi\epsilon\iota\rho$ means
hand, which explains the origin of the word chirality. Also some trees exhibit
a preference for a left-handed swirl and others for a right-handed swirl.
Snails are another such example. In the microscopic world, the biological
significance of a preferred handedness was discovered by Pasteur (1853) by
analyzing the effect of tartaric acid on polarized light.

Polarization is a property of transversal waves when the wave shows
oscillations perpendicular to the direction of propagation. This is different
for sound waves that are longitudinal. Unpolarized light consists of a
superposition of waves with all the different polarization orientations of the
wave plane. Using a polarizer, which is an optical filter with maximum
transmission for a particular wave plane, one can determine the orientation of
polarization. It turns out that natural sugar in solution has the property of
rotating the plane of polarization of polarized light in the right-handed
sense, so they are called dextrorotatory, denoted by $(+)$, while many amino
acids in solution rotate polarized light in the left-handed sense, denoted by
$(-)$. Pasteur (1853) also inspected the shapes of crystals of tartaric acid
under the microscope and found upon separating them that the two rotate
polarized light in opposite senses.

Figure 3.1:  An amino acid that is chiral whenever the residue R is different
from H. For example, when ${\rm R}={\rm CH}_{3}$, we have alanine, but when
${\rm R}={\rm H}$, the molecule is glycine, which is the same as its mirror
image, i.e., it is achiral. (Source:
https://chem.libretexts.org/@api/deki/files/19089/molecule.png?revision=1)

Handedness of biomolecules is primarily a consequence of the tetrahedral shape
of the of carbon compounds; see Figure 3.1. If each of the four bonds of the
carbon atom connect to a different group, its three dimensional structure
would be different from that of its mirror image. In the case of complex
molecules, there can be several carbon atoms that cause a violation of mirror
symmetry. Those carbon atoms are then called chiral centers. In the case of
tartaric acid (Figure 3.2), there are two chiral centers. There is then also
the possibility that only one of the two chiral centers is different. That
version is called meso-tartaric acid and it is achiral, i.e., it is
mirrorsymmetric.

There is no immediate connection between the handedness of molecules (Figure
3.2) and the handedness hidden in the structure of a crystal (Figure 3.3). In
fact, the association of a given chiral structure with left or right relies on
some convention. This also explains that there is nothing strange in having
right-handed sugars in our DNA and left-handed amino acids in our proteins.
Nevertheless, the very fact that something can occur in two possible forms
that are mirror images of one another is non-trivial and requires some
underlying structure that can also be subdivided into two opposite mirror
images of each other. It is therefore plausible that one of the two
handednesses of the l-tartaric acid molecule crystallizes into macroscopic
structures of one form, and the d-tartaric acid into its mirror image
(Derewenda, 2008). In the molecular context, these two forms are called
enantiomers.111 We must emphasize that the terminology in terms of
levorotatory and dextrorotatory is quite different from that in terms of d and
l. Levorotatory/dextrorotatory is the physical property for a compound to
induce the rotation of polarized light to the left/right. This property is
abbreviated $(-)$/$(+)$. By contrast, l/d refers to a structural property of a
molecule to denote its handedness, that is solely based on conventions. This
convention only applies to specific biomolecules, including amino acids and
sugars. This convention has been taken in such a way that all biogenic sugars
are d, and all biogenic amino acids are l. There is yet another terminology in
which R and S refer to a structural property denoting the handedness of a
given chiral carbon in a molecule. This is also based on a convention, which
applies to any chiral organic compound. This convention has been taken in such
a way that any chiral carbon can be assigned uniquely an R or S handedness
given a precise set of rules and is thus a drastically different convention
from l/d. For example, biogenic l-alanine is dextrorotatory $(+)$ and its
chiral carbon is of configuration S; biogenic l-serine is levorotatory $(-)$
and its chiral carbon is of configuration S; biogenic l-cysteine is
dextrorotatory $(+)$ and its chiral carbon is of configuration R. Regarding
sugars, d-glucose is dextrorotatory and d-fructose is levorotatory. Note also
that common table sugar (sucrose, i.e., a d-glucose–d-fructose dimer) is
dextrorotatory. If one hydrolyzes it, one obtains a 1:1 mixture of
d-glucose:l-fructose, corresponding to a mixture that is levorotatory, and
thus its common name of “inverted sugar”.

Figure 3.2:  Dextrorotatory (left) and levorotatory (right) tartaric acid.
Adapted from Sevin (2015).  Figure 3.3:  Original drawing from Pasteur’s
publication (Pasteur, 1922) showing dextrorotatory (left, denoted Fig. 1) and
levorotatory (right, denoted Fig. 4) tartaric acid. Adapted from Sevin (2015).

Interestingly, already back then, Pasteur made the statement that the
occurrence of handedness is a demarcating property between living and
nonliving matter; see Goldanskii & Kuz’min (1989). Particularly important is
Pasteur’s discovery of 1857 that certain unidentified microorganisms had a
considerable preference consuming $(+)$-tartaric acid over $(-)$-tartaric
acid; see the review articles by Gal (2008) and especially Sevin (2015). The
connection between a preferred handedness of biomolecules and living matter
was reinforced in a number of subsequent papers. The first important one was
by Frank (1953), who started his paper by saying “I am informed by my
colleague Professor W. Moore that there is still widely believed to be a
problem of explaining the original asymmetric synthesis giving rise to the
general optical activity of the chemical substances of living matter.” He then
proposed a model, which contained two key ingredients for producing a
systematic handedness: autocatalysis and mutual antagonism. Autocatalysis
means making more of itself. This is of course a governing principle of
biology, but it is meant here to be used at the molecular level during
polymerization, i.e., when long chains of shorter monomers are being assembled
into a long macromolecule. When each building block of the polymer has the
same chirality, one says that it is isotactic. Mutual antagonism, on the other
hand, can be interpreted as the tendency for a monomer of the wrong handedness
to spoil the polymerization, so that the polymer would no longer be isotactic.

The basic principle discovered by Frank has been governing many of the ideas
reflected in subsequent work in the field of homochirality. One such example
was the work of Fajszi & Czégé (1981), who also proposed a mathematical model
closely related to that of Frank. However, there are various other clues to
the question of homochirality on Earth. One is that there is handedness in one
of the four basic forces in nature, the weak force. We explain the details
below, but this discovery implies that certain properties of a chiral
molecule, for example the dissociation energy, can be different for the two
enantiomers. The energy difference is usually a very small fraction – below
$10^{-10}$ of the energy of the molecule itself; see Bonner (2000) for a
review. Because of the smallness, it is not obvious that this alone can be
responsible for achieving full homochirality. Thus, it is generally believed
that some amplification mechanism is always needed.

Figure 3.4:  Circular polarization measurements of the star-forming region
OMC-1 in the Orion constellation. Note that the circular polarization is
predominantly positive in the bulk of the molecular cloud. Courtesy of Bailey
et al. (1998).

An interesting astrobiological connection emerges when considering circularly
polarized light from astrophysical sources. This is light where the
polarization plane rotates with time or position. Star-forming regions in the
Orion constellation have been found to emit circularly polarized light
preferentially in only one of two possible senses; see Bailey et al. (1998);
Bailey (2001); see Figure 3.4 for an image of circular polarization in the
Orion molecular cloud (OMC). This is interesting because different enantiomers
can dissociate or degrade differently under the influence of circularly
polarized light. There is further support for this line of thought in that the
chirality of amino acids in space, for example in meteorites, is found to show
a slight preference for the levorotatory ones.

We mentioned already that the connection between chirality and the origin of
life goes back to an early suggestion by Pasteur. Another connection to
astrobiology arises when considering origins of life on other worlds. We will
return to this at the end when we discuss possible ways of assessing the
reality of extinct or extant life on Mars.

##  3.2 Enantiomeric cross inhibition: the need for homochirality

We mentioned already that the connection between the origin of homochirality
and the origin of life has been suspected since the early work of Pasteur.
This connection became more concrete with an important discovery of Joyce et
al. (1984). He performed experiments with polynucleotide templates, which
facilitate polymerization with the complementary monomers of the same
handedness.222Instead of polymerization, one sometimes talks about
polycondensation to emphasize the fact that polymerization implies the removal
of water in the reaction.

It was thought that polynucleotide templates of one handedness would direct
the pairing with monomers of the same handedness and therefore favor the
selection of nucleotides of the same chirality. Joyce et al. (1984) performed
experiments with polymers of dextrorotatory (d) cytosine (C) nucleobases,
poly(C${}_{\mbox{\sc d}}$), that are expected to pair with guanosine (G) mono-
nucleotides to form short strands, oligo(G${}_{\mbox{\sc d}}$), along the
poly(C${}_{\mbox{\sc d}}$). This was indeed the case and led to the formation
of up to 20 base pairs if the solution contained only monomers that are also
dextrorotatory; see Figure 3.5a. By contrast, when the solution contained only
levorotatory monomers, no polycondensation occurred; see Figure 3.5b.

Figure 3.5:  Chromatograms from the work of Joyce et al. (1984) showing
template-directed polycondensation of oligo(Gd) on poly(Cd) templates with d
mononucleotide (left panel), l mononucleotide (middle panel), and a racemic
mixture of d and l mononucleotide (right panel).

This was also expected, because base pairs with opposite handedness do not fit
together. The surprise came when using a racemic mixture of d and l mono-
nucleotides. A racemic mixture would indeed be expected under prebiotic
conditions. However, in that case there was no significant
polycondensation—not even with the d mononucleotides; see Figure 3.5c. Thus,
the idea of using template-directed polycondensation to select only one of two
handednesses did not work out. This phenomenon, which is known as enantiomeric
cross inhibition, turned therefore out to be a major problem for the RNA world
(Gilbert, 1986), unless there was a reason to expect that only monomers of one
handedness would be around. Joyce et al. (1984) wrote that “this inhibition
raises an important problem for many theories of the origin of life”. Bonner
(1991) credited Gol’danskii and Kuz’min in saying “that a biogenic scenario
for the origin of chiral purity was not viable even in principle, since
without preexisting chiral purity the selfreplication characteristic of living
matter could not occur.” This is where the discovery of the weak force comes
into play. It provides a reason why one particular handedness might be
preferred. This will be discussed next.

##  3.3 The weak force: non-mirrorsymmetry in nature

At the atomic level, there is the strong and the weak force. They are two of
the four fundamental forces in nature: gravity, the electromagnetic force, the
weak force, and the strong force; see the early review by Ulbricht (1975) in
the astrobiological context. The weak force is still rather strong compared
with gravity ($10^{24}$ times stronger than gravity), but weak compared with
the electromagnetic force ($10^{11}$ times weaker). The weak force is
responsible for the decay of free neutrons, whose half-time is only about 10
minutes. The neutron (n) decays then into a proton (p) and an electron (e).
This, as well as the reverse process (electron capture), occur also in the
nuclei of atoms, for example in the decay of radioactive potassium-40 into
calcium-40 and argon-40, where the half-time is $1.25\,{\rm Gyr}$. While there
is significant astrobiological significance in this, for example for dating
rocks,333 Measuring the argon inclusions in solidified rocks is the basis for
determining the age of rocks. The potassium-40 isotope constitutes only 0.01%
of naturally occurring potassium. Its half-time is $1.25\,{\rm Gyr}$, making
it ideal for geochronology. The two decay reactions are  ${}^{40}_{19}{\rm
K}\longrightarrow\;^{40}_{20}{\rm Ca}:\quad{\rm n}\to\;{\rm p}+{\rm
e}+\bar{\nu}_{\rm e}\quad\mbox{($\beta$ decay)},$ (3.1) ${}^{40}_{19}{\rm
K}\longrightarrow\;^{40}_{18}{\rm Ar}:\quad{\rm p}+{\rm e}\to\;{\rm
n}+\nu_{\rm e}\quad\mbox{(electron capture)}.$ (3.2) The latter reaction is
responsible for the argon in the atmospheres of Earth and Mars.  we are here
concerned with the fact that the electrons from the decay of neutrons are
always left-handed. This means that the spin of the electron is anti-aligned
with its momentum; see Figure 3.6. At low energies, however, the spin can flip
relative to the momentum, so the handedness of electrons is predominantly a
high energy phenomenon.

Figure 3.6:  Illustration of lepton helicity. The momentum $\bm{p}$ is a polar
vector, while the spin $\bm{S}$ is an axial vector, so their dot product is a
pseudoscalar, so it changes its sign when inspected in a mirror. The electron
from the beta decay of a neutron has $\bm{p}\cdot\bm{S}<0$ and is referred to
as left-handed.

The fact that electrons produced by $\beta$ decay are chiral is remarkable,
because it means that our physical world is, at least in some respects,
different from its mirror image. This goes back to a remarkable discovery by
Lee & Yang (1956), which earned them the Nobel Prize in Physics of 1957 “for
their penetrating investigation of the so-called parity laws which has led to
important discoveries regarding the elementary particles.”

The connection between the chirality of electrons and that of biomolecules is
not immediately evident. There are two different ways of establishing a
connection between the handedness imposed by the electroweak force and the
handedness in the biomolecules. One is through the fact that the
bremsstrahlung emission from chiral electrons rotating around magnetic field
lines is circularly polarized with a sense of polarization that depends on the
chirality of the electrons. This implies that the sense of polarization from
bremsstrahlung is always negative and that this radiation destroys
preferentially right-handed amino acids through photolysis. This was found by
Goldhaber et al. (1957) and McVoy (1957) in back-to-back papers in the
Physical Review almost immediately after the influential paper by Lee & Yang
(1956). If the idea that circularly polarized light can affect the stability
and selection of biomolecules is to make any sense, one should be able to
discover polarized light in nature. Interestingly, star-forming regions of
OMC-1 in the Orion constellation have indeed been found to emit right-handed
circularly polarized light (Bailey et al., 1998), supporting this basic idea;
see Bailey (2001) for a discussion of the astrobiological implications.
However, the circular polarization observed by Bailey et al. (1998) occurred
at near-infrared wavelengths and is not related to the mechanism of Goldhaber
et al. (1957) and McVoy (1957), who considered circularly polarized
bremsstrahlung. Bailey et al. (1998) argued that the observed circular
polarization is caused by Mie scattering of unpolarized light, but this
mechanism is unrelated to the weak force. It is therefore conceivable that
also left-handed circularly polarized light could have been produced in the
opposite direction.

Instead of relying on starlight, there is yet another possibility. Muons, like
electrons, belong to the group of fermions that tend to have a certain
handedness. Muons are about 200 times more massive than electrons and can
therefore be more effective in producing strongly circularly polarized
radiation. Muons occur in the cosmic radiation on Earth. They are only
produced when an energetic cosmic particle hits the Earth’s atmosphere and
produces a muon shower. For this reason, the muons in the cosmic radiation can
play a significant role in affecting the chirality of biomolecules Globus &
Blandford (2020). Unlike the observed circular polarization from the OMC-1 in
the Orion constellation, the sense of circular polarization from this
mechanism is connected with the weak force and therefore, just like in the
case of bremsstrahlung, only one of the two senses are possible, giving rise
to the preferential destruction of right-handed amino acids.

There is another completely different connection between biomolecules and the
weak force. Quantum-mechanical calculations have shown that the dissociation
energies for d and l molecules are slightly different (Hegstrom, 1984;
Hegstrom et al., 1980; Mason & Tranter, 1984). Therefore, the d and l amino
acids in a racemic mixture will degrade at different rates, which leads to an
excess of l amino acids.

##  3.4 Chiral amino acids in meteorites

Amino acids have been found in some meteorites (Engel & Macko, 1997). Two
particular meteorites are often discussed in connection with the enantiomeric
excess of amino acids: the Murray and the Murchison meteorites (Pizzarello &
Cronin, 2000). Those are carbonaceous chondrites, which means that they are
carbon-rich. They are also rich in organics, as was superficially evidenced by
the smell reported by initial eyewitnesses of the Murchison meteorite.
Interestingly, Table 1.5 of Rothery et al. (2008) lists 18 different amino
acids that have been found not only in the Murchison meteorite, but also in
the Miller–Urey experiment (Miller, 1953). Twelve of them are not found in
proteins on Earth. This is interesting, because it suggests that those amino
acids were indeed originally present in the meteorite and could not have come
from contamination by life after the meteorite landed on Earth. Those amino
acids that are found on Earth include glycine, alanine, valine, proline,
aspartic acid, and glutamic acid

The sense of the enantiomeric excess is the same in the two meteorites,
corresponding to levorotatory amino acids, but the amount is different
(Pizzarello & Cronin, 2000). In addition, there is the possibility that the
enantiomeric excess may be caused by terrestrial contamination (Bada, 1995).
But, as emphasized above, this would only apply to the six amino acids that
are also found on Earth. In particular, those amino acids that have the
clearest enantiomeric excess are also those that are most vulnerable to
contamination; see Ehrenfreund et al. (2001) for a discussion of terrestrial
contaminants in connection with the carbonaceous chondrites Orgueil and Ivuna.
They are of the type CI (I for Ivuna) and are extremely fragile and therefore
susceptible to terrestrial weathering. In Orgueil, alanine was found to be
racemic and was argued to be abiotic in origin (Ehrenfreund et al., 2001).
They could not, however, support the suggestion of terrestrial contamination
with corresponding soil samples. Incidently, the Orgueil meteorite is also
known for a famous contamination hoax; see Anders et al. (1964), who discusses
the paper by Cloez (1864) claiming the existence of life on the meteoritic
parent body a few weeks after Pasteur’s famous lecture to the French Academy
on the spontaneous generation of life.

Among the possible causes for the enantiomeric excess of meteoritic amino
acids, there is the aforementioned effect of circularly polarized starlight.
Circularly polarized ultraviolet light could have preferentially destroyed one
of the two chiralities through photolysis (Zeldovich et al., 1977). The
experiments of Bonner et al. (1981) with a d l mixture of leucine showed that
right-handed circular polarized light leads to a preferential destruction of d
leucine, while left-handed circular polarized leads to a preferential
destruction of l leucine; see also (Meierhenrich & Thiemann, 2004) for recent
experiments. To explain the systematic l excess of amino acids on Earth, one
would need the protosolar nebula to be irradiated by right-handed polarized
light. Indeed, the star-forming region OMC-1 has been found to emit right-
handed circularly polarized light, supporting this basic idea (Bailey et al.,
1998; Bailey, 2001; Boyd et al., 2018). However, as discussed above, also
left-handed circularly polarized light could have been produced in the
opposite direction. Therefore, any systematic l excess of amino acids caused
by this mechanism would have been by chance.

The enantiomeric excess found in some amino acids is at most around 1–2%. This
would be too small to avoid the problem reported by Joyce et al. (1984). So,
even if there is an external effect producing a systematic enantiomeric
excess, we always need an amplification mechanism. Therefore, we discuss next
the Frank mechanism and move then to some variants of it that avoid either
autocatalysis or enantiomeric cross inhibition. We begin by explaining first
the basic idea.

##  3.5 The basic idea behind the Frank mechanism

The essence of the mechanism of Frank (1953) is the combination of two
ingredients operating in a substrate: catalysis of molecules for their own
production and “anticatalysis” that corresponds to some antagonism or
deleterious effect. He even talks about “poisoning” one of the two enantiomers
out of existence. In fact, he called his simple mathematical model a “life
model”, suggesting already back then that he was thinking of them as being
processes acting at the moment when the first life emerged.

The essence of Frank’s model is perhaps best explained graphically. For this
purpose, it is most instructive to begin with the deleterious effect by
assuming that an equal amount of d and l enantiomers eliminate each other in
each reaction step. This is illustrated in Figure 3.7, where we indicate the
amount of d enantiomers with blue bars and the amount of l enantiomers with
red bars.

Figure 3.7:  Sketch showing the effect of enantiomeric cross inhibition only.
Red and blue bars indicate opposite enantiomers, with the blue one being
initially in the majority by one “unit”, the separation between subsequent
bars. The gray columns indicate the total amounts, which is 5 units for the
column with blue bars and 4 units for the column with red bars. In the end, in
step 4, only one unit of the enantiomer marked with blue bars survives.

We see that, in the end, only d is left (see the blue bars), but the amount is
very small, namely just as big as the initial difference by which one of the
two enantiomers exceeded the other. This is why we also need autocatalysis.
Autocatalysis is a process that is not enantioselective, i.e., it works the
same way for the d and l enantiomers. This is demonstrated by stretching out
the columns by a factor such that the highest column always retains the
original height; see Figure 3.8. It is instructive to quantify here the
enantiomeric excess (e.e.) as the ratio of the difference to the sum of the
concentrations of right- and left-handed compounds, i.e.,

$\mbox{e.e.}=\frac{[D]-[L]}{[D]+[L]}$ (3.3)

At each each, the value of e.e. in Figs. 3.7 and 3.8 is the same:
$1/(5+4)=1/9$ initially, then $1/(4+3)=1/7$, $1/(3+2)=1/5$, $1/(2+1)=1/3$, and
finally $1/1=1$.

Figure 3.8:  Similar to Figure 3.7, but in each reaction step, the separations
between subsequent bars has been stretched by a certain factor such that the
column with the blue bars retains the same height. The stretching emulates the
effect of autocatalysis. In the end, again only the enantiomers marked with
blue bars survive, but now, because of the stretching, the amount is no longer
small.

Frank’s paper was mathematical, which may have been a reason why it was not
widely recognized in the biology community at the time. In fact, it was not
quoted in the work of Joyce et al. (1984), who discovered enantiomeric cross
inhibition in the context of nucleotides. It was only with the paper of
Sandars (2003) that Frank’s mutual antagonism was identified with enantiomeric
cross inhibition. It was clear from Frank’s work that, as long as both
reactions, autocatalysis and antagonism, remain active, the racemic state is
unstable and there will be a bifurcation into a chiral state with an excess of
either d or l enantiomers; see also Sandars (2005), where enantiomeric cross
inhibition was no longer regarded as a problem, but as an essential ingredient
in achieving full homochirality.

In the Frank mechanism, there must be at least a very small initial imbalance
which will then be amplified. However, this is not a problem because, even if
we tried to construct a purely racemic mixture in the laboratory, there will
always remain a tiny imbalance. This is just for the same reasons that in a
cup of blueberries we will hardly ever have exactly the same number
twice.444If you take a cup of blueberries, for example, the exact number
varies between 65 and 70
(https://www.howmuchisin.com/produce_converters/blueberries), so we must
always expect there to be a small imbalance in the number if we say we have an
equal amount of d and l enantiomers. Mathematically, this imbalance grows with
the square root of the number of molecules (or blueberries) and would be about
$\pm 10^{12}$ for one mole with ${\cal N}=6\times 10^{23}$ molecules (or $\pm
8$ for $65$ blueberries); the fractional imbalance is $1/\sqrt{\cal
N}=10^{-12}$ in one mole (or $12\%$ for $65$ blueberries).

Given that the racemic state is unstable, the enantiomeric excess, as defined
in Eq. (3.3), will grow exponentially in time and it does therefore not matter
how small the initial imbalance in the concentrations of d and l was. To
demonstrate this more clearly, we use here a figure of Brandenburg et al.
(2007), who considered the model of Plasson et al. (2004), which we discuss
later in more detail in Section 3.10. This model also has the property that
the racemic state is unstable and that the system evolves toward one of the
two homochiral states.

Figure 3.9:  Probability distribution of the initial enantiomeric excess
(e.e.) for racemic mixtures with $10^{6}$ and $10^{12}$ molecules together
with the resulting evolution of e.e., both in logarithmic and linear
representations. The dashed lines give a gaussian fit to the distribution
function. Adapted from Brandenburg et al. (2007).

In the following, we discuss an ensemble of solutions of the model of Plasson
et al. (2004) with different realizations or initial states, which consisted
of a racemic mixture of equally many d and l enantiomers. Figure 3.9 shows
that one always obtains a fully homochiral state, but in about 50% of the
cases (or in 50% of the realizations of the same experiment), one obtains
eventually a state with either just d enantiomers, and in the other 50% of the
cases or realizations, one with only l enantiomers. When we talk about
different cases or realizations, we must realize that the genesis of life on
Earth is just one such realization. Another one may have occurred on Mars, or
in the atmosphere of Venus, or elsewhere in the Galaxy. Of course, there is
also the possibility of multiple geneses on Earth alone, with certain
lifeforms being either completely or partially wiped out (Davies & Lineweaver,
2005). The latter case may be particularly interesting in models where we
allow for chemical evolution in models with spatial extent, which will also be
discussed later in Section 3.9.

##  3.6 Evidence for autocatalysis

Unlike the process of enantiomeric cross inhibition, where we have referred to
the experiments of Joyce et al. (1984), the actual evidence for autocatalysis
is poor. In fact, there is only the classical reaction of Soai et al. (1995)
that exhibits autocatalysis and can lead to a finite enantiomeric excess; see
Gehring et al. (2010) and Athavale et al. (2020) for more recent work
clarifying the implications of the Soai reaction. However, the basic idea of
autocatalysis remains plausible, especially since the discovery by Guerrier-
Takada & Altman (1984) and Cech (1986) that RNA molecules can exhibit
autocatalytic functionality. This was a very important discovery that earned
Sidney Altman and Thomas R. Cech the Nobel Prize in Chemistry in 1989 “for
their discovery of catalytic properties of RNA.” It is this mechanism that is
at the heart of the idea of an RNA world (Gilbert, 1986).

It is important to realize that the existing evidence for autocatalysis is
irrelevant from an astrobiological viewpoint. This is particularly clear in
view of the fact that the Soai reaction requires zinc alkoxides as an
additional crucial catalyst. Those compounds are not generally believed to
play a role on the early Earth.

Autocatalysis in the sense of making more of itself is obviously a basic
principle of life, but this is already at a rather complex and not at the
level of individual molecules. It is therefore possible that autocatalysis
does not play a significant role and that it is rather the process of network
catalysis (Plasson, 2015), i.e., the combined action of different molecules
that lead to the desired appearance of what is in the end equivalent to
autocatalysis. We will return to this in Section 3.10, when we discuss a
particular sequence of reactions that, in the end, have the effect of
autocatalysis, even though autocatalysis is not present in any individual
reaction.

##  3.7 The effect of an external chiral influence

In the beginning of this review, we have discussed extensively the possibility
of a systematic bias resulting eventually from the fact that the weak force
introduces a preference of one of two handednesses through one or several
possible effects. Those would always favor l amino acids and d sugars. On the
other hand, we have now seen that the Frank mechanism can result in full
homochirality of either chirality. Does this mean that the bias introduced by
the weak force is unimportant? Maybe not quite. It depends on how strong the
external influence is in comparison with the speed of autocatalysis, which
determines the rate of the instability. This was first discussed in the work
of Kondepudi & Nelson (1983, 1985) in papers that appeared just at the time as
that of Joyce et al. (1984), but, at the time, neither of those authors
mentioned the work of Frank.

The paper by Kondepudi & Nelson (1983, 1985) was in principle quite general
and therefore applicable to other symmetry breaking instabilities. In essence,
the effect of the bias is that it makes the bifurcation asymmetric. A
symmetric bifurcation is one where the enantiomeric excess (positive or
negative) departs away from strictly zero as some bifurcation parameter
increases. Sandars (2003) identified this bifurcation parameter with the
fidelity of the autocatalytic process, which measures the probability with
which the catalytic process does indeed facilitate the polymerization with
monomers of the same chirality instead of the opposite one. The fidelity $f$
is unity (zero) when the autocatalytic process always (never) produces
polymerization with the same handedness.

Figure 3.10:  Bifurcation diagrams showing a slight preference for positive
enantiomeric excess (e.e., here denoted by $\eta$). The left panel has been
adapted from Brandenburg et al. (2005), where the bifurcation begins for a
fidelity $f$ that is clearly below the otherwise critical value of $f=0.5$.
(The dashed line denotes the unstable solution.) The right panel has been
adapted from Brandenburg (2019), who considered a stochastic model where
$f=0.2$ was assumed.

In Figure 3.10 we show a bifurcation diagram from the work of Brandenburg et
al. (2005), where we see that for all values of the fidelity $f$, the solution
with positive enantiomeric excess ($\eta$) is stable. For
$\eta\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr>\cr\sim\cr}}}}0.5$,
the solution with negative $\eta$ is also stable, but to reach this solution,
the initial fluctuations must be large enough. The complete bifurcation
diagram also contains an unstable solution, which corresponds to the watershed
between the two stable branches. In the left hand plot of Figure 3.10, it is
shown as a dashed line. Similar diagrams have also appeared in the works of
Kondepudi & Nelson (1983) and later in the review of Avetisov et al. (1991).

##  3.8 Polymerization model of Sandars (2003)

Looking at the chromatographs of Joyce et al. (1984), we see that the ultimate
goal is to assemble long polymers. For this reason, Sandars (2003) developed a
polymerization model for d and l nucleotides, where he also allowed for
enantiomeric cross inhibition. In his model, monomers of the d and l forms are
being produced at rates, $Q_{D}$ and $Q_{L}$, respectively, that are
proportional to same reaction rate $k_{C}$ and the concentration of some
substrate $[S]$, i.e.,

$Q_{D}=k_{C}[S]\Big{\\{}{\textstyle{1\over 2}}(1+f)C_{D}+{\textstyle{1\over
2}}(1-f)C_{L}+C_{0D}\Big{\\}},$ (3.4)
$Q_{L}=k_{C}[S]\Big{\\{}{\textstyle{1\over 2}}(1+f)C_{L}+{\textstyle{1\over
2}}(1-f)C_{D}+C_{0L}\Big{\\}},$ (3.5)

where $0\leq f\leq 1$ is the fidelity, $C_{L}$ and $C_{D}$ are parameters
describing the global handedness of the system [the concentrations of the
longest possible chains of left- and right-handed polymers for Sandars (2003)
and quantities proportional to the masses of all polymers of the d and l forms
for Brandenburg et al. (2005)]. These parameters are introduced in such a way
that for $f>0$ in Eqs. (3.4) and (3.5), $Q_{D}$ increases with $C_{D}$, and
$Q_{L}$ increases with $C_{L}$. The parameters $C_{0D}$ and $C_{0L}$ allow for
the possibility of non-catalytic production of left- and right-handed
monomers. They can be different from zero when there is an external bias or
external influence. When $C_{0D}=C_{0L}=0$, Eqs. (3.4) and (3.5) show that
$Q_{D}=k_{C}[S]C_{D}$ and $Q_{L}=k_{C}[S]C_{L}$ when $f=1$, while
$Q_{D}=Q_{L}=k_{C}[S](C_{D}+C_{L})/2$ when $f=0$.

Sandars (2003) assumed that the catalytic effect depends on the concentrations
of the longest possible chains of left and right handed polymers. Brandenburg
et al. (2005) adopted a similar model, but assumed that $C_{D}$ and $C_{L}$ to
be proportional to the masses of all polymers of the d and l forms,
respectively. This allowed them to extend the model to much longer polymers
without needing to wait for the longest one to appear before autocatalysis
became possible at all.

The full set of reactions included in the model of Sandars (2003) is (for
$n\geq 2$)

$\displaystyle L_{n}+L_{1}$ $\displaystyle\stackrel{{\scriptstyle
2k_{S}~{}}}{{\longrightarrow}}$ $\displaystyle L_{n+1},$ (3.6) $\displaystyle
L_{n}+D_{1}$ $\displaystyle\stackrel{{\scriptstyle
2k_{I}~{}}}{{\longrightarrow}}$ $\displaystyle L_{n}D_{1},$ (3.7)
$\displaystyle L_{1}+L_{n}D_{1}$ $\displaystyle\stackrel{{\scriptstyle
k_{S}~{}}}{{\longrightarrow}}$ $\displaystyle L_{n+1}D_{1},$ (3.8)
$\displaystyle D_{1}+L_{n}D_{1}$ $\displaystyle\stackrel{{\scriptstyle
k_{I}~{}}}{{\longrightarrow}}$ $\displaystyle D_{1}L_{n}D_{1},$ (3.9)

where $k_{S}$ and $k_{I}$ are suitably chosen reaction rates for symmetric
autocatalysis and (non-symmetric) inhibition, respectively. For all four
equations we have the complementary reactions obtained by exchanging
$L\rightleftarrows D$. The polymerization starts from a large but limited set
of monomers that all begin to develop longer polymers. Because the number of
monomers was limited, the theoretically obtained chromatograms show a
characteristic wave-like motion with increasing time; see Figure 3.11.

Figure 3.11:  Normalized concentration $[L_{n}]$ versus $n$ showing a wave-
like evolution of an initial Gaussian profile (solid line). All later times
are shown as dashed lines, except for the last time, which is shown as a long-
dashed line. Adapted from Brandenburg et al. (2005).

Particularly important is of course the case where monomers of both
chiralities exist. In that case, the result depends on the fidelity of the
autocatalytic reactions; see Figure 3.12 for such a result. We see that longer
polymers can only be produced when the fidelity is relatively high. The lack
of sufficient fidelity therefore explains the limited length of polymers found
in the work of Joyce et al. (1984).

Figure 3.12:  $[L_{n}]$ (left) and $[L_{n}D]$ (right) of equilibrium solutions
for different values of $f$. For $f=1$ we have $[L_{n}D]=0$, which cannot be
seen in the logarithmic representation. Adapted from Brandenburg et al.
(2005).

##  3.9 Spatiotemporal chirality dynamics

In all the chemical reactions discussed so far, the assumption was made that
the system is well mixed. This means that the concentrations $[D_{n}]$,
$[D_{n}L]$, etc, are the same everywhere. On larger length scales, this
assumption must eventually break down. Even on the scale of alkaline
hydrothermal vents, where many scientists place the origin of life (Russell,
2006) the relevant chemical reactions would take place within small semiporous
cells. It is then conceivable that similar reactions take place in neighboring
compartments that would be formed by the sulfurous precipitants from these
vents. Russell (2006) draws here an analysis to the chemical gardens that
would allow for a growing arrangement of new compartments, which could act as
primitive cells and would, in principle, allow for Darwinian evolution as
these chemical reactions propagate from one layer of compartments to the next;
see also Russell et al. (2014) and Barge et al. (2017, 2019) for more recent
developments. In each of these compartments, strong spatial gradients and
$10^{8}$-fold concentration enhancements can be achieved through thermal
convective flows when the aspect ratio of the compartment is sufficiently
large (Baaske et al., 2007). This setup can also lead to oscillations, which
can locally lead to exponential replication of nuclei acids, analogous to the
polymerase chain reaction (Braun & Libchaber, 2002).

In the scenario described above, we can no longer talk about a well mixed
system. Therefore, the concentrations must be regarded as function not only of
time, but also of space. Because the chemistry in neighboring compartments is
loosely coupled by diffusion terms, there would be spatio-temporal evolution.
In that case, the chemical reaction equations attain a spatial diffusion term.
The resulting system of equations is usually referred to as reaction–diffusion
equations. Such models, but for only one instead of several species, have
frequently been employed in modeling the dynamics of diseases such as the
black death (Noble, 1974) or rabies (Källén et al., 1985; Murray et al.,
1986). It has also been used to model the spreading of the novel coronavirus,
where the total number of cases was found to follow a quadratic or piecewise
quadratic growth behavior (Brandenburg, 2020).

To address the question of homochirality in an extended system, Brandenburg &
Multamäki (2004) employed a similar approach, but with two or multiple
species. Multiple species occur when we invoke polymers of different length
and composition of different species for the d and l forms. They found that a
given species tends to spread through front propagation. It turned out that,
once two populations of opposite chirality meet, the front can no longer
propagate and the evolution comes to a halt. This result was first obtained in
a one-dimensional model, where the concentrations of d and l depend on just
one spatial coordinate $x$ and on time $t$. The result is shown in Figure
3.13, for the evolution of short polymers. These are all regarded as separate
species. The initial condition consists of a small number of monomers of the l
form at one position (at $x/\lambda=-0.1$ in Figure 3.13, where $\lambda$ is
the length of the domain) and a three times larger number of monomers of the d
form at another position (at $x/\lambda=+0.1$). The theoretical chromatograms
are stacked next to each other for each $x$ position. We see that in both
positions, longer polymers are produced, indicated by the yellow-reddish
colors. The initially threefold larger number of d monomers is insignificant,
because the growth is exponentially fast soon saturates at the same level as
that for the l monomers, when the polymers have reached their maximum size. At
the same time, polymers of the same handedness can still be produced by
diffusion to the neighboring positions. This leads to a propagation front.
However, when polymers of opposite chirality emerge at neighboring positions,
the front stops (here at $x\approx 0$), while on the other two sides, the
reaction fronts still diffuse further outward.

Figure 3.13:  Color scale plots of $[D_{n}]$ and $[L_{n}]$ after $0.8$
diffusion times as a function of position $x$ and polymer length $n$. In
$0<x/\lambda<0.25$, only d polymers exist (left) and no l polymers at all
(right), while in $-0.25<x/\lambda<0$ it is the other way around. Adapted from
Brandenburg & Multamäki (2004).

More interesting dynamics is possible when the system is two-dimensional,
corresponding to different locations on the Earth’s surface. In that case, the
fronts between regions with monomers and polymers of opposite handedness can
be curved. It turned out that then the fronts are never precisely straight and
can therefore still propagate. Interestingly, the propagation is always in the
direction of maximum curvature. This result has also been obtained by Gleiser
& Walker (2012). This then implies that a closed circular front will always
shrink and never expand; see left Figure 3.14. The speed of shrinking depends
just on the number of individual closed fronts. Each time a closed front
merges with another one to form a single one, the speed decreases; see the
right panel of Figure 3.14. In particular, this means that, even if, say, the
l enantiomers were initially in the majority, but in such a way that they
would be enclosed in an island surrounded by enantiomers of the opposite
handedness, the enantiomeric excess would develop toward the handedness that
was present on the periphery of the domain; see Figure 3.15 for such an
illustration with the model of Plasson et al. (2004).

Figure 3.14:  Left: Fractional concentration of one chirality versus position
($x$ and $y$) at four different times $t$, normalized by the diffusivity
$\kappa$ per total surface area $\lambda^{2}$, so $t\kappa/\lambda^{2}$ is
nondimensional. In this numerical simulation,
$\kappa/(\lambda^{2}\lambda_{0})=2\times 10^{-4}$, and the resolution was
$1024^{2}$ mesh points. The number of disconnected regions decreases from 4 in
the first plot to 3, 2, and 1. Right: Evolution of enantiomeric excess $\eta$
for the model shown in the left. The inset shows the normalized slope. Note
the four distinct regimes with progressively decreasing slope. Adapted from
Brandenburg & Multamäki (2004).

Figure 3.15:  Left: Shrinking of an initially large patch of molecules of the
d form surrounded by molecules of the l form. Right: The resulting
enantiomeric excess (e.e.) versus time $t$, scaled with the activation rate
$a$, so $at$ is nondimensional. Note that it was initially positive, but
reaches later complete homochirality with $\eta=-1$.

Figure 3.16:  Sketch illustrating that densely packed discs inside the
periphery of a circle differ in their number from those outside the periphery
by just 3. This result is independent of the total number; compare the left
and right illustrations with 60 and 120 discs, respectively. As time goes on,
pairs of red and blue discs get eliminated and the circle shrinks, because the
number of discs inside the periphery is slightly smaller (by three) than the
number of discs outside the periphery. The smallness of the difference in the
numbers on the inner and outer peripheries is the reason for the shrinking of
the circle slowing down.

The reason for this particular propagation direction is quite simple. Imagine
that we place d and l molecules around a circular front, then the number of
molecules on the inner front is would be one less than the number of molecules
on the outer front; see Figure 3.16 for an illustration.555 To understand why
the difference in the number of molecules between the outer and inner circles
is always just three, let us imagine the molecules being represented by little
discs of radius $r$ on the periphery of a circle of radius $R$. The
circumferences of the outer and inner peripheries are $2\pi(R\pm r)=2\pi
R\pm\pi r$ for the upper and lower signs, respectively. The difference is
therefore $2\pi r\approx 3d$, where $d=2r$ is the diameter of each disc. The
difference in the number of discs is there for three.

##  3.10 Recycling Frank: the peptide model of Plasson et al.

We said already in Section 3.6 that autocatalysis may not be a particularly
evident process on the early Earth. For that reason, Plasson et al. (2004)
devised a completely different mechanism that they advertised as “recycling
Frank”. It is based on the combination of the following four important
reactions: activation (A), polymerization (P), epimerization (E), and
depolymerization (D). So the resulting model is also referred to as the APED
model. A variant of this model was studied by Konstantinov & Konstantinova
(2018).

It is important to emphasize that there is no explicit autocatalytic reaction.
However, the combined sequence of reactions (Brandenburg et al., 2007)

$D+L\stackrel{{\scriptstyle
a}}{{\longrightarrow}}D^{*}+L\stackrel{{\scriptstyle
p}}{{\longrightarrow}}DL\stackrel{{\scriptstyle
e}}{{\longrightarrow}}LL\stackrel{{\scriptstyle h}}{{\longrightarrow}}L+L,$
(3.10) $L+D\stackrel{{\scriptstyle
a}}{{\longrightarrow}}L^{*}+D\stackrel{{\scriptstyle
p}}{{\longrightarrow}}LD\stackrel{{\scriptstyle
e}}{{\longrightarrow}}DD\stackrel{{\scriptstyle h}}{{\longrightarrow}}D+D.$
(3.11)

does effectively result in an autocatalytic reaction, but it is not a direct
one. First of all, it requires an activation step, indicated by asterisk, a
polymerization step (with the rate constant $p$), an epimerization step (with
the rate constant $e$), and finally a depolymerization step (with the rate
constant $h$). Because the autocatalysis in indirect, this sequence of steps
can therefore be regarded as a simple example of a network catalysis (Plasson,
2015; Hochberg et al., 2017).

##  3.11 Fluctuations instead of autocatalysis or enantiomeric cross-
inhibition

During the last decade, there has been some increased interest in the role of
fluctuations; see a recent review by Walker (2017). Fluctuations can play
important roles in diluted systems, in which the number of molecules is small.
In such case, rate equations no longer provide a suitable description of the
relevant kinetics when the system is dilute and reactions are rare (Gillespie,
1977; Toxvaerd, 2014). In that case, a stochastic approach must be adopted.
This may be relevant to the work of Toxvaerd (2013), where homochirality has
been found without apparent autocatalysis or enantiomeric cross-inhibition.
Instead of solving rate equations, as discussed in the previous sections, one
solves stochastic equations. This means that at each reaction step, the state
of the system changes, but with a reaction that is taken to depend on chance
with a certain probability. The system is then described by vector
$\bm{q}=(n_{A},\,n_{D},\,n_{L})$, where $n_{A}$ denotes the numbers of achiral
molecules and $n_{D}$ and $n_{L}$ denotes the number of molecules of the d and
l forms, respectively. In the model of Brandenburg (2019), seven different
reactions were considered, each with a certain probability. Not all those
seven reactions need to be possible in a certain experiment, so the
probability for some reactions can be zero. Applying a single reaction step
with enantiomeric cross inhibition implies

$(n_{A},\,n_{D},\,n_{L})\stackrel{{\scriptstyle
k_{\times}~{}}}{{\longrightarrow}}(n_{A}+2,\,n_{D}-1,\,n_{L}-1),$ (3.12)

i.e., the numbers of $D$ and $L$ get reduced by one, and that of $A$ increases
by two. We can also include spontaneous deracemization reactions, i.e.,

$\displaystyle(n_{A},\,n_{D},\,n_{L})\stackrel{{\scriptstyle
k_{+}~{}}}{{\longrightarrow}}(n_{A}-1,\,n_{D}+1,\,n_{L}),$ (3.13)
$\displaystyle(n_{A},\,n_{D},\,n_{L})\stackrel{{\scriptstyle
k_{+}~{}}}{{\longrightarrow}}(n_{A}-1,\,n_{D},\,n_{L}+1).$ (3.14)

To model different reaction rates, the different reactions must happen with
different probabilities. This is done by taking at each reaction step a random
number between zero and one. Suppose we want to model enantiomeric cross
inhibition together with spontaneous deracemization, then the probability that
the first reaction happens is proportional to $k_{\times}$, and the
probability that one of the other two reactions in Eqs. (3.14) and (3.14)
occurs is proportional to $k_{+}/2$. If we also allow for the possibility that
nothing happens (probability proportional to $k_{0}$), then our scheme with
$\bm{q}\to\bm{q}+\Delta\bm{q}$ is as follows:

$\displaystyle\mbox{if}\quad 0\leq r<r_{1}\equiv k_{\times}$
$\displaystyle\mbox{then $\Delta\bm{q}=(2,-1,-1)$},$ (3.15)
$\displaystyle\mbox{if}\quad r_{1}\leq r<r_{2}\equiv r_{1}+k_{+}/2$
$\displaystyle\mbox{then $\Delta\bm{q}=(-1,1,0)$},$ (3.16)
$\displaystyle\mbox{if}\quad r_{2}\leq r<r_{3}\equiv r_{2}+k_{+}/2$
$\displaystyle\mbox{then $\Delta\bm{q}=(-1,0,1)$},$ (3.17)
$\displaystyle\mbox{if}\quad r_{3}\leq r<1$ $\displaystyle\mbox{then
$\Delta\bm{q}=\bm{0}$ (no reaction)}.$ (3.18)

Note that $k_{\times}+k_{+}+k_{0}=1$ is here assumed. This particular
experiment was referred to as experiment III in Brandenburg (2019), where
$k_{0}=0$ was assumed. As he varied $k_{\times}$, $k_{+}$ was assumed to vary
correspondingly such that $k_{+}=1-k_{\times}$. The results of this experiment
are similar to those with spontaneous deracemization replaced by
autocatalysis, which is referred to as experiment I in Figure 3.17. Here, the
autocatalysis rate is varied such that $k_{\rm C}=1-k_{\times}$. This is the
standard Frank model, but for a diluted system, while model III is close to
that of Sugimori et al. (2008, 2009), who were the first to find a transition
to full homochirality even without autocatalysis. Next, in experiment II,
there is autocatalysis, but no enantiomeric cross inhibition and just
spontaneous racemization instead. This type of model was first considered by
Jafarpour et al. (2015, 2017). The transition to full homochirality was
originally though impossible in such a model (Stich et al., 2016).

Figure 3.17:  Bifurcation diagrams of $\langle|\eta|\rangle$ (black) and
$\langle A\rangle$ (red) for $N=3000$ (solid lines) and $N=300$ (dotted lines)
as a function of parameters for models I, II, III, and IV. Adapted from
Brandenburg (2019).

In a comparative study, all these processes were studied within a single
unified model. In Figure 3.17 we show the results of the four different
experiments. We mentioned already experiments I–III. In experiment IV, by
comparison, there is just racemization and deracemization, but neither
autocatalysis nor enantiomeric cross inhibition. In that case, the average of
the modulus of the enantiomeric excess, $\langle|\eta|\rangle$, no longer
reaches unity, but levels off at about $0.5$ when
$k_{+}\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr>\cr\sim\cr}}}}0.4$.
We also see from the red lines that the achiral compounds ($A$) get depleted
in favor of producing chiral ones either of the d or the l form.

##  3.12 Chirality from a Martian Labeled Release experiment

Back in 1976, when the Viking I and II landers visited the Chryse Planitia and
Utopia Planitia regions, respectively, many of the things we now know about
Mars were still unclear. In particular, the existence of water on Mars was
still very much an open question. Nevertheless, one was relatively optimistic
at the time. Both landers came with advanced experiments on board to look for
life. One of the experiments, the Labeled Release (LR) experiment, was
actually successful (Levin & Straat, 1976, 1977), but another experiment never
detected any organics, which was decisive enough to conclude that no life was
detected after all (Klein et al., 1976).

The idea behind the LR experiment is simple: take Martian soil, mix it with
water and organics as nutrients, and see whether a metabolic reaction occurs
that decomposes the nutrients and produces a gaseous waste product, for
example methane or carbon dioxide; see the recent account by Levin & Straat
(2016), where detailed tests with various terrestrial soils were presented.
The carbon atoms of the nutrients were labelled with carbon-14 isotopes, a
technique commonly used in medicine, which allows one to trace those labeled
carbon atoms by their radioactivity. To identify the gaseous waste product,
one simply measured the level of radioactivity. Control experiments with
sterilized soil showed that only fresh Martian soil produced a reaction. The
Viking laboratories were flexible enough to perform additional experiments
with lower sterilization temperatures. The critical temperature below which no
sterilization occurred was found to be around $50^{\circ}$. Those temperatures
would appear reasonable for Martian cryophiles, but are generally too low for
sterilization on Earth. The experiment was tested in various deserts on Earth
and it was able to detect metabolism at measurable levels.

It is only since 2012 that organics were detected on the Martian surface by
the Curiosity rover; see Voosen (2018) for a popular account. We also know
that organics get quickly destroyed by perchlorates, in particular KClO4,
which were discovered on the Martian surface by the Phoenix lander in 2008
(Hecht et al., 2009). Such processes could potentially result in reactions
found with the LR experiment (Quinn et al., 2005), but it remains puzzling why
a critical sterilization temperature of $50^{\circ}$ was found, and not much
higher, for example. Thus, while an explanation in terms of abiotic processes
has not been fully conclusive (Valdivia-Silva, 2012), the explanation that
life was actually detected might seem more straightforward (Levin & Straat,
2016). However, as already noted by Carl Sagan, “the more extraordinary the
claim, the more extraordinarily well-tested the evidence must be. The person
making the extraordinary claim has the burden of proving to the experts at
large that his or her belief has more validity than the one almost everyone
else accepts.” In any case, it seems justified to repeat this experiment to
clarify the phenomenon that the Viking landers discovered back in 1976; see
also the recent paper by Carrier et al. (2020).

Figure 3.18:  Metabolic consumption of ${\sc d}{\;}$-glucose (filled symbols)
and ${\sc l}{\;}$-glucose (open symbols) and by (1) Saccharomyces cerevisiae,
(2) Penicilium expansum, (3) E. coli, (4) Micrococcus luteus, (5)
Natronobacterium sp., and (6) Halostagnicola sp. Adapted from Sun et al.
(2009).

Given that the experiment is relatively simple and can detect life under harsh
conditions on Earth, it would be interesting to repeat some variants of it in
the future. One such variant would be to allow for the detection of
handedness. This would constitute a more conclusive signature of life than
just the discovery of some metabolism. Such an experiment can be done by using
chiral nutrients, which goes back to the old findings of Pasteur of 1857 that
certain microorganisms had a preference for consuming $(+)$-tartaric acid over
$(-)$-tartaric acid; see the reviews by Gal (2008) and Sevin (2015).

In Figure 3.18 we show the result of recent experiments by Sun et al. (2009)
using different types of eukarya, bacteria, and archaea, which were given
either d sugars or l sugars. In most of the cases there was a clear preference
in the microbes taking up the naturally occurring d sugars compared with the
synthetically produced l sugars. Subsequent work showed that the specificity
for some microbes is low and that some of those can use sugars of the opposite
chirality also (Moazeni et al., 2010). Although the dependence on the type of
nutrients has not yet been studied in detail, it may be important to allow for
a broad range of different ones in an attempt to account for such ambiguities.

##  3.13 Conclusions

Louis Pasteur was well ahead of his time when he identified the biological
role of chirality in living matter. Particular remarkable is his realization
that, during fermentation, the metabolic uptake of nutrients of opposite
chiralities is different. To understand why this experiment was not put in the
context of extraterrestrial life detection, we have to realize that in those
years, it was not uncommon to think of extraterrestrial life on Mars. When the
astronomer Herschel (1784) discovered seasons on Mars, he wrote in his paper
in the Philosophical Transactions of the Royal Society that this “planet has a
considerable but moderate atmosphere so that its inhabitants probably enjoy a
situation in many respects similar to ours.” So, not just the existence of
life, but the existence of intelligent of life on Mars was commonly expected.
This only changed in 1964, when Mariner 4 returned the first flyby pictures of
Mars, which suggested that any life there would probably only be of microbial
nature. But that Vikings 1 and 2 would not even find any organics on Mars was
such a shock to many that the search for life in the Universe appeared fairly
hopeless, and Mars exploration was put on hold for the next two decades. This
all changed since the turn of the century with the discovery of extremophiles
on Earth and the realization that terrestrial life has existed since the time
that stable continents existed. Gradually, with the conclusive detection of
water on Mars, the search for extinct or extant life on Mars restarted, and
Pasteur’s discovery of different metabolic uptakes of d and l nutrients may
finally turn into an actual Martian experiment. As shown by Sun et al. (2009),
this property can be used to detect the presence of homochirality through in
situ experiments. Homochirality can also be detected through remote sensing by
looking for circular polarization. This approach has been persued by Patty et
al. (2019), who found negative or left-handed circularly polarized light
emitted from terrestrial plant life at about $680\,$nm; see also Avnir (2021)
for a recent review. To what extent this technique can be used as a biomarker
still needs to be seen, but it is amazing to see once more how Pasteur’s early
discoveries have shaped some important aspects of astrobiology.

While homochirality remains a property that is strongly associated with
life—or at least some chemical process that keeps the system far from
equilibrium—it is not clear whether we should expect it to have the same or
the opposite handedness as an Earth (Bada, 1996). To answer this question, one
would need to have more realistic models with meaningful estimates for the
concentrations of suitable chemicals in some protocells. This would allow for
an estimate of the level of fluctuations in relation to the strengths of the
small but systematic effects resulting from the weak force. It would be the
only way of guaranteeing that each genesis of life always produces the same
chirality. But for now, we should be satisfied if one could find (and
understand) any type of an extraterrestrial metabolic process that works
differently for nutrients of d and l forms. Thus, the discussed possibilities
would need to be put on a quantitatively meaningfully basis. And if it is not
life, it certainly is interesting enough to deserve serious attention!

## Acknowledgements

I am grateful to the two referees for many useful comments and suggestions
that have led to improvements of the manuscript. In particular, I wish to
acknowledge Raphaël’s clarifications on the differences in the terminology in
terms of levorotatory and dextrorotatory on the one hand and that in terms of
d and l on the other. This work was supported in part through the Swedish
Research Council, grant 2019-04234. I acknowledge the allocation of computing
resources provided by the Swedish National Allocations Committee at the Center
for Parallel Computers at the Royal Institute of Technology in Stockholm.

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