and gives tokens to all places it has arcs pointing to. The number of tokens
transferred depends on the weightings on the individual arcs (by default, they
are 1). In place/transition nets, there is no explicit concept of time, only
of events which take place in sequences.

Once a system has been converted to a place/transition net representation, a
well-developed set of theorems and algorithms can be applied to decide general
behavioural possibilities of the network (e.g. see ?, ?). For example, it can
be decided whether the network is live (all transitions are able to contribute
to the net behaviour, forever), reversible (any state can always be returned
to) and how the places are bounded (the maximum number of tokens that each
place will hold during execution of the net). Also, invariant features of the
network can be identified from its static structure, such as sets of places
which tend to conserve total tokens (‘P-Invariants’) and sets of transitions
whose firing does not change the marking (‘T-Invariants’). These invariants
often have interpretations relevant for the system understudy. For example
when metabolic networks are rendered as place/transition nets, P-invariants
correspond to pools of conserved substrates, and T-invariants can give insight
into steady state behaviour.

### Petri Nets as Structured Descriptions of Model
Equations

Since their conception as basic place/transition nets, petri nets have
diversified into a ‘zoo’ of many different petri net classes, where extra
elements and arcs have been added to increase their modelling power for
various different applications. Rather than being applied as a theoretical
tool to give insight into network behaviour, one popular application of petri
nets is simply as a visual structured description of a dynamical model. A
model can be mocked-up by drawing a petri net and then easily converted into
either a continuous or stochastic description ready for numerical
execution333The dynamical model usually needs to be expressible as a set of
first order derivatives. (?, ?).

In particular, a special class of stochastic petri nets (called XSPN) has been
developed for turning vague notions of biochemical pathways into exact models,
which can then be simulated and investigated quantitatively, allowing
comparison to wet lab data (?, ?). Indeed, this class of petri net can be used
to specify the protocell model used in this thesis (see Section 3.3.2). Then,
the protocell model can be executed on a publicly available simulation
platform called Snoopy (?, ?, ?). However, the latter simulation platform is
not capable of modelling any of the protocell morphological transformations
that require modification of the model equation structure (e.g. division)
because the petri net simulated is static. Throughout a simulation, the
network topology is fixed and the boundary parameters are fixed.

### Extending Petri Nets to Handle Structural Change in Dynamical Systems

Petri nets are actually good candidates for modelling constructive dynamical
systems with changing equations and/or boundary conditions. Even though a
petri net representation of a dynamical model contains the same amount of
information as the differential equations of that model, the visual format of
the petri net makes the relationships between processes and state variables
much more obvious444In the same way, a nested LISP program and its tree
representation have the same information content, but the tree structure is
easier to work with when modifying the program e.g. when combining programs in
genetic programming.. As such, petri nets provide a convenient vehicle for
altering the structure of a dynamical model, and then for consistently
carrying these changes through to formulate new overall evolution equations.

Figure A.1 shows how the conventional petri net paradigm can be combined with
an unconventional part, in order to obtain a model which can change not only
in state, but also in structure, as it executes. The idea of this architecture
is that a petri net is simulated in a ‘running’ phase (Fig. A.1a) where the
token marking changes and the structure is fixed, but additionally the petri
net may pass transiently into a ‘remodelling’ phase (Fig. A.1b). In the
remodelling phase, four types of more drastic modifications can be made to the
network, alone or in combinations: (i) large, discontinuous jumps in the state
space, (ii) changes to the boundary parameters of the system, (iii) changes to
the connectivity of the system (leaving the degrees of freedom unchanged) or
(iv) expansion or contraction of the degrees of freedom of the system.

Figure A.1:  A Two Phase Petri Net Architecture for Modelling Constructive
Dynamical Systems. See text for discussion.

In order to pass between the running and remodelling phases, a new type of
transition can be introduced into a petri net, called a ‘remodel’ transition
(not shown in Fig. A.1). This type of transition fires immediately on becoming
enabled, representing that the current state variables have fallen into some
critical range or ratios, and the current model structure no longer reflects
the underlying system being modelled. Rather than moving tokens, a remodel
transition instead executes a sequence of instructions which describe how the
petri net network should be remodelled. Then, the modified network is returned
to the running phase. Many such remodelling events may happen during the
course of a simulation.

Crucially, the instructions executed by the remodel transitions hold
information about the underlying objects the dynamical system is describing
(see the discussion of the work of Fontana and Buss in Section 2.2.3). When
the state of the system gets to the existence limit of the current objects the
dynamical system is describing, the remodel transitions fire to modify the
underlying objects in the appropriate way, creating a new dynamical system.

### Worked Example: Petri Net Model for Growing and Dividing Protocells

As part of this thesis, the novel two phase petri net architecture described
above was implemented in C and subsequently used to perform a diversity of
numerical protocell simulations. This program (called Flow) accepts stochastic
petri nets in the XSPN format referred to above. Additionally, the petri nets
can be specified as modules and connected together, allowing initially
complicated or repetitive network designs to be achieved (i.e. grids of
connected systems).

For a brief overview, Fig. A.2 shows an example petri net capable of modelling
a basic vesicle division cycle. The vesicle grows by internally synthesising
membrane lipid from a precursor that diffuses through the membrane (Fig.
A.2a). The petri net consists of five modules, four of which are connected
together to make the vesicle (Fig. A.2c, blue lines). Table A.1 details the
initial token marking for the petri net, and the rate formula for each of the
transitions. The petri net has three types of transition: ‘normal’ transitions
(open squares), ‘immediate’ transitions (black rectangles) and remodel
transitions (red rectangles), a special type of immediate transition. Normal
transitions have a stochastic propensity (rate) and are fired with the
standard Gillespie algorithm. Immediate transitions fire as soon as they
become enabled555If many immediate transitions are simultaneously enabled,
they are fired at random., and they are useful for updating meta-information
about the network when certain conditions hold. For example, the immediate
transitions IT1 and IT2 are responsible for monitoring the direction of the
concentration gradient of precursor S; similarly, immediate transitions
IT3-IT6 are responsible for monitoring the surface area to volume ratio of the
vesicle. The remodel transitions (red rectangles) execute instructions to make
more drastic changes to the petri network. Dotted lines in Fig. A.2, drawn
from places to transitions in the petri net, are ‘modifier’ edges denoting
that a place influences the rate of a transition, but the place has no tokens
consumed when that transition fires.

In this division example, it does not make sense to follow both daughter
vesicles after a fission event because the surrounding environment is a
reservoir, and the daughter vesicles will not be able to interact with each
other. Therefore, only one daughter is followed, and this means that the
degrees of freedom of the system remain the same. The remodel transition
handling division RT_DIVIDE executes the following instructions when enabled
by a token in the DIVIDE place:

  1. 1)

Calculate how many buffer molecules B are in the vesicle. B = I \- S \- L

  2. 2)

Make a discontinuous state change:

Half the vesicle contents: tokens on S = S / 2; B = B / 2; L = L / 2

Half the membrane surface area: tokens on $L_{\mu}$ = $L_{\mu}$ / 2 (some
randomness could be added to the division process if desired)

  3. 3)

Set the total number of solute molecules666State variable I is not actually a
true state variable of the vesicle. It is included however, because it allows
the easy addition of processes influencing (and influenced by) the volume of
the vesicle. Without this pseudo-state variable, building a protocell from
petri net modules would be much more brittle. inside the vesicle I = S \+ L \+
B

  4. 4)

Calculate $\Phi$ for the new daughter vesicle, and put a token in the TENSION
place if $\Phi<1$, or in the DEFLATED place if $\Phi\geq 1$.

  5. 5)

Run the modified petri network (which now represents the followed daughter
vesicle).

Figure A.2:  Petri Net Model for a Basic Protocell Division Cycle. (a) A
vesicle with internal lipid synthesis (b) exhibits the dynamic behaviour of a
decreasing division cycle when (c) modelled as an XSPN petri net with added
remodel transitions. Table A.1 below gives the formula for the petri net
transitions.

Initial Petri Net Marking
---
Starting with 100nm diameter spherical vesicle
$V_{(sph)}=\frac{4}{3}\pi 50^{3}\times 10^{-24}\text{litres}$,
$S_{\mu(sph)}=4\pi 50^{2}\times 10^{-16}\text{dm}^{2}$
Buffer $B$ inside vesicle is set to give initial isotonic condition.
State Variable (Place) |  Tokens
$S$ |  0
$L=l_{\cal E}N_{A}V_{(sph)}$ |  16
$B=(s_{\cal E}+b_{\cal E})N_{A}V_{(sph)}$ |  66217
$I=S+L+B$ |  66233
$L_{\mu}=\frac{2S_{\mu(sph)}}{\alpha_{L}}$ |  209440
IN |  1
OUT |  0
BURST |  0
TENSION |  0
DEFLATED |  1
DIVIDE |  0
TIMERON |  1
TICK |  0
Transition Firing Rates
Normal Transitions |  Stochastic Propensity
T1 |  $\text{OUT}\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{S}}S_{\mu}\frac{(S/\Omega)-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}s_{\cal E}}}{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\lambda}}$
T2 |  $\text{IN}\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{S}}S_{\mu}\frac{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}s_{\cal E}}-(S/\Omega)}{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\lambda}}$
T3 |  ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}k_{r}}S$
T4 |  ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}k_{out}}L_{\mu}$
T5 |  ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}k_{in}}S_{\mu}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}l_{\cal E}}$
T6 |  ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}k_{out}}L_{\mu}$
T7 |  ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}k_{in}}S_{\mu}\frac{L}{\Omega}$
T8 |  $\text{TIMERON}\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}k_{tick}}$
Immediate Transitions |  Fire when True
IT1 |  $\text{OUT}\times({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}s_{\cal E}}\geq\frac{S}{\Omega})$
IT2 |  $\text{IN}\times({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}s_{\cal E}}<\frac{S}{\Omega})$
IT3 |  $\Phi<1-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\epsilon}$
IT4 |  $\Phi\geq 1$
IT5 |  $\Phi<1$
IT6 |  $\Phi>(1-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta})\sqrt[3]{2}$
Remodel Transitions |  Execute Instructions when True
RT_PERTURB |  $\text{TICK}>{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}T}$
RT_DIVIDE |  $\text{DIVIDE}==1$
RT_BURST |  $\text{BURST}==1$

Table A.1: Initial Marking and Transition Propensities for Petri Net of Fig.
A.2. Boundary parameters of the system are shown in blue. To prevent the
transition formulas being a mess of symbols, ‘intermediate’ variables are
used. These are relationships between the true state variables, defined as:
total internal molecules
$I=S+L+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}B}$,
scaled volume
$\Omega={I}/({{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}s_{\cal
E}+b_{\cal E}}})$, surface area
$S_{\mu}=L_{\mu}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\alpha_{L}}/2$,
and reduced surface
$\Phi={S_{\mu}}/{\sqrt[3]{36\pi\left(\Omega/{N_{A}}\right)^{2}}}$.

In other words, RT_DIVIDE resets the state of the parent vesicle to that of
the first daughter, and then recommences the simulation. Figure A.2b shows
that a decreasing vesicle division cycle will result from this system, because
the surface area of the vesicle membrane grows, but the vesicle volume does
not.

Module 5 does not form part of the vesicle petri net, but rather triggers a
perturbation to the environment of the vesicle after an approximate time has
elapsed. In this case, this perturbation is envisaged to be a decrease in the
external buffer concentration $b_{\cal E}$, such that the vesicle experiences
an influx of water, growing in volume. After a specific number of tokens have
accumulated on place TICK, the remodel transition RT_PERTURB fires, executing
the following instructions:

  1. 1)

Make a boundary conditions change:

Set the parameter $b_{\cal E}$, representing the buffer concentration in the
environment, to a lower value.

  2. 2)

Make a discontinuous state change:

Reset the tokens on TICK to 0.

Remove the token from TIMERON, to disable any further perturbations to the
environment.

  3. 3)

Run the modified petri network.

The effect of this event is shown as the green bar on Fig. A.2b: the vesicle
instantly increases in volume. Finally, remodel transition RT_BURST will just
contain the instruction to stop the simulation if the vesicle bursts. If the
scenario were different, e.g. a population of vesicles existed in a finite
environment, this transition could be used to remove the modules of the
vesicle that burst, and add a new module representing a flat bilayer, in their
place.

In summary, representing a model protocell system as a flat petri net
structure is advantageous for a number of reasons. Firstly, the model is a
transparent white box: all of its workings are clear. It is straightforward to
publish the precise model used to produce simulation data, so that others can
reproduce findings exactly. Secondly, new model features can be added by the
user by changing the petri net, and are not dependent on (bug prone)
modifications to the underlying software. Finally, a small and efficient
algorithm can be developed for simulating a two phase petri net architecture;
an algorithm that will be robust and bug-free regardless of simulating small
systems or large, complicated networks.

##  Appendix B Lattice Model of Self-Assembly in Surfactant-Water-Oil Mixtures

This appendix contains more technical details on the lattice model of
amphiphilic self-assembly reported in Section 4.1.2. The lattice model was a
‘spin’ model of the Larson type (?, ?; see ?, ? for a review) and implemented
in two dimensions on a 100x100 lattice.

### Spin Models to Capture Universal Properties of
Complex Phenomena

Lattice spin models are a class of model typically used in statistical physics
to investigate the equilibrium properties of solids and liquids. A famous
example of a spin model is the Ising model (?, ?; ?, ?). In this model, a
solid piece of magnetic material, say iron, is represented as a regular
lattice of spins. Each spin represents an iron atom pointing either upwards
(spin of +1) or downwards (spin of -1) and locally adjacent spins have the
tendency to align with each other. The lattice is connected to a heat bath
which provides a thermal noise, counteracting the alignment of spins. Even
though the Ising model is an extremely simplified model of real piece of iron,
it is nevertheless able to reproduce the phase transition in the magnetisation
of an iron solid when it is heated through a critical temperature (and to
quite remarkable accuracy, see ?, ?). In the Ising model, at low temperatures,
there is a global alignment of spins (i.e. the solid is magnetised) up to a
critical temperature. At the critical temperature there is the sudden onset of
wide fluctuations in the spin alignments, and past the critical temperature
all spins are essentially randomised due to excessive thermal noise (i.e. the
solid becomes de-magnetised). Therefore, although the Ising model is a brutal
abstraction of the atomic structure and interactions of a particular magnetic
solid, there is a sense in which the model has captured something _universal_
about the way magnetisation in general works.

Spin models of amphiphilic fluids follow a similar formalisation to the Ising
model and are also are intended to capture universal features – this time
about the way self-assembly works in ternary fluid mixtures of amphiphiles,
water and oil.

### General Description of Larson Model Evolution

The Larson model of amphiphilic fluids is based on a square lattice of binary
spins which have only nearest neighbour and diagonally nearest neighbour
interactions.111That is, the Moore neighbourhood. The four diagonally nearest
neighbours are treated as being at the same interaction distance as the four
nearest neighbours. The lattice is typically of two or three dimensions with
periodic boundary conditions in all directions. The Larson model uses spin to
denote hydrophobicity. In the basic model, a +1 spin signifies the presence of
a hydrophilic ‘water-loving’ moiety, and a -1 spin the presence of a
hydrophobic ‘oil-loving’ moiety. The lattice is completely filled, with up to
three type of molecules: water molecules $W$ are represented as single +1
spins, oil hydrocarbons $O$ as single -1 spins, and amphiphiles, denoted
$H_{i}T_{j}$, are connected chains of $i$ hydrophilic head $H$ spins (+1)
followed by $j$ hydrophobic tail $T$ spins (-1). Amphiphile chains move as
self-avoiding walks with the constraint that they cannot be broken, nor cross
themselves or other neighbouring chains. Being part of a still liquid, the
molecules and amphiphile chains on the lattice are free to move around into
new configurations, but the number of molecules of each type is always
conserved.222Conversely in the Ising model (reviewed next), the atoms are
fixed in place, and the total number of up and down spins is not conserved.

The lattice has a total energy in any particular configuration, depending on
what spins are next to each other. As ? (?) writes, ‘the lattice model really
is simply a way of describing an ensemble of surfaces that can break and tear
but that possess an energy of deformation.’ (p1592). The hydrophobic effect is
modelled by giving all pairs of adjacent spins an appropriate local
interaction energy: like spins (hydrophobic-hydrophobic pairs, or hydrophilic-
hydrophilic pairs) decrease the lattice energy, whereas unlike spins
(hydrophobic-hydrophilic pairs) increase the lattice energy. At equilibrium,
closed physical systems always seek their lowest energy configuration: in the
case of the lattice, total energy is minimised when there exists a minimal
number of unlike spins in contact.

The lattice is started with a random distribution of oil, water and
amphiphiles, which represents a high energy non-equilibrium configuration.
This initial condition must be transformed into a series of lattice
configurations likely at equilibrium. From statistical physics, it is known
that when the lattice has equilibrated to some temperature $T$, the
fluctuations in the total energy of the configurations appearing should be
distributed according to the Boltzmann distribution. A special monte carlo
algorithm called the Metropolis-Hastings algorithm (see ?, ? for an excellent
pedagogical review) is used to re-arrange the oil, water and amphiphilic
chains on the lattice to accomplish this goal. The M-H algorithm drives the
system into the region of lattice configuration space where successive lattice
configurations have energy distributed according to the Boltzmann distribution
for temperature $T$.333Technically, the Metropolis-Hastings algorithm
constructs a markov chain of configurations which is stable on the Boltzmann
distribution. The M-H algorithm belongs to the class of Markov Chain Monte
Carlo algorithms. Generally, these algorithms are able to find and return
samples (termed ‘important’ samples) from regions of multidimensional
probability distributions where there is the most ‘mass’ (probability). When
in this region, the M-H algorithm is producing important samples of likely
lattice configurations for temperature $T$. These monte carlo samples can be
subsequently used to compute reliable equilibrium averages for certain
properties of the amphiphilic fluid on the lattice (e.g. if micelles are
formed, their number and size distributions can be calculated).

In practice, to ensure proper lattice equilibration, the lattice temperature
should be gradually decreased from a high value to the final temperature $T$.
This process is called simulated annealing (?, ?) and ensures that the system
does not stray too far-from-equilibrium. If the system is not gradually cooled
but, rather, is ‘quenched’ to a low temperature rapidly, non-optimal lattice
configurations can sometimes become frozen into the system.

### Lattice Energies

Interaction energies for different pairs of adjacent molecules are given as a
matrix and specified in terms of a dimensionless parameter $\omega$.444Simply
put, $\omega$ represents both the temperature and interaction energies of the
system. Defined as $\omega=\epsilon/kT$ it corresponds to the lowest common
denominator of the interaction energies in Joules, divided by (the Boltzmann
constant multiplied by the absolute Kelvin temperature of the system).
$\omega$ is used in place of using absolute interaction energies in Joules and
absolute temperatures in Kelvin, because it is only the dimensionless ratio of
energy to temperature that is important in the evolution of the model. For the
simple Larson model:

|  $W$ or $H$ |  $O$ or $T$
---|---|---
$W$ or $H$ | $-\omega$ | $\omega$
$O$ or $T$ | $\omega$ | $-\omega$
Table B.1:

The matrix is symmetric about the diagonal because the order of the adjacent
molecules does not matter (WO always has the same energy as OW). The total
energy of the lattice is described by the so-called Hamiltonian equation,
simply

${\cal
H}=\sum_{<ij>}E_{ij}N_{ij}=\left(E_{OO}N_{OO}+E_{WW}N_{WW}+E_{WO}N_{WO}\right)$
(B.1)

where $E_{ij}$ is the interaction energy between molecules of type $i$ and $j$
(drawn from the matrix above), and $N_{ij}$ is the number of times that a pair
of $i$ and $j$ type molecules are adjacent to each other on the lattice.

### Algorithm for Larson Model Evolution

  1. 1)

Initialise a random lattice configuration, completely filling the lattice with
water and oil molecules, and amphiphile chains. The concentration of each
‘species’ is defined as the amount of lattice area occupied by that species.

  2. 2)

Start the system at a high temperature, i.e. with $\omega$ close to 0.

  3. 3)

Update the lattice to the next configuration:

(i) Propose a move. Randomly choose to move either a single oil or water
molecule, or a whole amphiphile chain. Oil and water molecules move by
exchanging with adjacent molecules. Amphiphile chains move through a series a
transformations. A common chain movement is a snake-like movement whereby the
entire chain slithers one unit along its own path555Chain transformations
generally just need to be able to move chains into varied conformations whilst
keeping the system ergodic..

(ii) Calculate the energy change associated with the move ${\triangle{\cal
H}}$ from the Hamiltonian B.1.

(iii) Accept the proposed move with probability:

$P(accept)=min\left\\{1,e^{-\Delta H}\right\\}$ (B.2)

Therefore, all moves which lower the lattice energy are automatically
accepted, whereas moves which raise the energy are accepted with a probability
that exponentially decreases with the increase in energy.

  4. 4)

Repeat from step 3 until the total lattice energy has stopped changing,
signifying that the ‘burn-in’ period is over.

  5. 5)

Start the next annealing phase. Cool the system temperature a small amount by
increasing $\omega$ a small amount toward its target value, and repeat again
from step 3. Generally, as the temperature lowers, the more time the lattice
needs to equilibrate, and the longer the annealing phase required.

  6. 6)

When the target temperature is reached, lattice configurations generated by
the M-H algorithm can be used for the purposes of estimating desired expected
values.

In the model as implemented, the Step 3 above was performed $7\times 10^{7}$
times and the lattice temperature was decreased to $\omega=0.5$ from
$\omega=0.07$ over 7 annealing phases. Each annealing phase was twice the
length of the previous one.

#### A Note on Moving Amphiphile Chains

The problem of moving an arbitrary length amphiphile chain on the lattice
without breaking the chain, or crossing other chains, presented a particular
challenge (even in two dimensions). In this regard, two checks were helpful to
employ:

  1. 1)

Moore consistency. After a move, all adjacent beads of a chain must be one
Moore distance apart. If not, the chain is broken (illegal).

  2. 2)

Cross consistency. After a move, all diagonal sections of a chain must be
tested and found not to cross another chain segment. This is because on a
square lattice, chains can potentially cross each other (or even themselves)
when moving diagonally.

##  Appendix C Two Validity Tests for Abstract Elementary Reaction Sets

Theoretical protocell models often include a proto-metabolic network specified
in abstract terms, with chemical transformations symbolised as letters, e.g.
$A+B\rightarrow C$. A useful exercise is to develop validity tests for such
abstracted reaction sets, to ensure that they do not negate basic physical
laws. Such validity tests also permit the automatic generation of random
metabolisms.

This appendix proposes two validity tests which can be applied to abstract
sets of elementary reactions, one based on atom number conservation and
another on free energy changes. A reaction set passing both tests can be said
to be valid on basic conservative mass and energy grounds. The primary use of
the tests is to give a way to rule out implausible reaction mechanisms. The
two tests are dependent only on the stoichiometry of the reactions (i.e. the
reaction rates are not involved) and are applicable regardless of whether the
individual reactions in a reaction set are considered reversible or
irreversible, or a combination of both. Table C.1 gives examples of elementary
reaction sets passing zero, one or both tests.

Related work. Test 1 presented here seems to accomplish the same as, albeit in
a different way, the test for conservative chemical reaction networks
presented by ? (?). Test 2 has also been performed in the literature
indirectly by e.g. ? (?), but in the latter case, not in the form of an
explicit validity test. Moving towards more realistic chemistry, the work of ?
(?) is relevant, as they describe a computer algorithm that generates all the
possible multi-step elementary reaction sequences that could underlie a
particular overall reaction, given the list of reactants, intermediate and
product molecules/ions to be involved in the reaction. Working with real
chemistry, these authors are able to use more constraints. The work in this
appendix is generally related to the stoichiometric checking of chemical
kinetic models, the interested reader is directed to ? (?).

### Validity Test 1: Possibility for Atom Number
Conservation

During the course of a chemical reaction, the law of conservation of mass
states that the quantity of each atomic element does not change; the atoms
only become re-arranged into new molecules. In the case of real chemical
equations, simple elemental balancing or ‘atom counting’ can be used to ensure
that no reaction creates nor destroys mass.

In the case of abstracted chemical equations, similar balancing can still be
employed if the chemical species have labels which reflect the atomic groups
they consist of, e.g. binary polymers $AAB+ABB\rightarrow AABABB$. Here, it is
clear how the atom groups have re-arranged in each reaction.

However, if the abstracted chemical equations do not make explicit the atomic
constituents of each species, e.g. $A+B\rightarrow C$, then group balancing
cannot be performed. Nevertheless, conservation of the total _atom number_ in
the system can still be verified (a weaker condition) by rewriting the set of
reactions as a set of linear simultaneous equations. To do this, every
reaction arrow (regardless if irreversible, or bi-directional) is re-written
as an equals sign, and the species letters become variables signifying the
number of atoms in a molecule of that type. Each linear equation stipulates
that the _total number_ of atoms on the left and right hand sides has to be
equal.

If at least one positive non-zero solution can be found to the simultaneous
equations, then this indicates that all species can be assigned a positive
atom number, such that no sequence of reactions will neither create nor
destroy atoms. The abstract reaction set has a possibility of being
implemented in real chemistry. On the other hand, if a set of elementary
reactions fails atom number conservation, then the set is _certainly invalid_
: somewhere in the chemical transformations is hidden the destruction or
spontaneous creation of atoms.

Usefully, the solutions to the simultaneous equations indicate which are the
largest and smallest molecules in the system.

Box C1: Validating conservation of atoms Reaction set H in Table C.1 | Solve simultaneous equations
---|---
$2w\rightarrow y$ | $2w-y=0$
$y+w\rightarrow 2x$ | $y+w-2x=0$
$x+w\rightarrow y+z$ | $x+w-y-z=0$
$2z\rightarrow w$ | $2z-w=0$
| where $x>0$, $y>0$, $z>0$, $w>0$

### Validity Test 2: Possibility for All Free Energy
Changes as Negative

Without having to perform experiments, the standard Gibbs free energy change
of a reaction $\Delta G^{o}$ is a good predictor of whether or not that
reaction will take place spontaneously. $\Delta G^{o}$ is defined as the total
standard Gibbs free energy of formation of the products, minus the total
standard Gibbs free energy of formation of the reactants:

$\Delta G^{o}=\sum_{p}^{\text{products}}\Delta
G_{f}^{o}(p)-\sum_{r}^{\text{reactants}}\Delta G_{f}^{o}(r)$ (C.1)

If negative111i.e. the total standard Gibbs free energy of formation of the
products is more negative than that of the reactants, then the reaction
process is spontaneous under standard conditions (1 atm pressure, 298K
temperature, 1 M solute concentration). Standard Gibbs free energy of
formation is typically listed in tables for many substances.

The standard Gibbs free energy change can be used as an approximate assessment
as to whether a set of abstract elementary reactions is viable from an
energetics perspective. The assessment is only approximate, because (i)
thermodynamic spontaneity is only assessed relative to standard conditions
(whereas reaction conditions could be non-standard, for example), and because
(ii) even if a reaction is spontaneous, which in thermodynamic terminology
means that it _can_ occur, this does not mean that the reaction necessarily
does occur at a kinetically appreciable rate. Nevertheless, a basic energetics
test is useful to apply to abstract elementary reaction sets, since it rules
out perpetual motion cycles like $A\rightarrow B\rightarrow C\rightarrow A$,
which have no problems passing validity test 1.

To test for free energy adherence, an elementary reaction set is be
transformed into a set of linear inequalities, with each reaction arrow
pointing in the spontaneous direction, re-written as a greater-than $>$
symbol, and with each species letter representing the zero or negative
standard Gibbs free energy of formation for that species.

If a set of solutions exists to this set of inequalities (i.e., a solution
simplex is defined in the negative orthant of N-dimensional space, where N is
the number of species), then this indicates that all species can be assigned a
zero or negative standard Gibbs free energy of formation, such that all
reactions can proceed spontaneously under standard conditions. The reaction
set has a possibility of being implemented in real chemistry.

The solutions to the inequalities, for valid reaction systems, dictate a
general energy ordering of molecules in the reaction network, but this
ordering is generally flexible and there are many energy combinations which
are possible.

Box C2: Validating standard free energy change Reaction set H in Table C.1 | Solve inequalities
---|---
$2w\rightarrow y$ | $2w-y>0$
$y+w\rightarrow 2x$ | $y+w-2x>0$
$x+w\rightarrow y+z$ | $x+w-y-z>0$
$2z\rightarrow w$ | $2z-w>0$
| $x\leq 0$
| $y\leq 0$
| $z\leq 0$
| $w\leq 0$

Reaction Set A |  Reaction Set B |  Reaction Set C |  Reaction Set D
---|---|---|---
(invalid) |  (invalid) |  (invalid) |  (invalid)
1\. $2z\rightarrow x+y$ |  1\. $z\rightarrow y+w$ |  1\. $2x\rightarrow y+z$ |  1\. $2w\rightarrow 2x$
2\. $y\rightarrow z+w$ |  2\. $y+z\rightarrow x$ |  2\. $y+w\rightarrow 2x$ |  2\. $2z\rightarrow y$
3\. $x\rightarrow y+w$ |  3\. $y+w\rightarrow 2x$ |  3\. $x+y\rightarrow 2w$ |  3\. $2x\rightarrow 2w$
4\. $2w\rightarrow y+z$ |  4\. $y+w\rightarrow z$ |  4\. $z\rightarrow x$ |  4\. $2y\rightarrow w$
valid atom conservation solutions?
NO |  NO |  YES |  YES
$w=0$, $x=0$, $y=0$, $z=0$ |  $w=\frac{3z}{2}$, $x=\frac{z}{2}$, y = $-\frac{z}{2}$ |  $w=z$, $x=z$, $y=z$ |  $w=4z$, $x=4z$, $y=2z$
valid free energy solutions?
NO |  NO |  NO |  NO
Sub inequality from (3) into (1) gives $2z>(y+w)+y$. Sub inequality from (2) into previous answer gives $2z>2z+3w$, which is impossible. |  From (1) $z>y+w$, but (4) asserts the opposite $y+w>z$, which cannot be true also. |  Inequalities for (2) and (3) can be written $y>2x-w$, $y>2w-x$ respectively. Sub inequality from (4) into (1) gives $2x>y+x$, or $x>y$. Sub $x$ for $y$ in (2) and (3) gives $y>2y-w$, $y>2w-y$ which reduce to the unsolvable pair $w>y$, $y>w$. |  From (1) $2w>2x$, but (3) asserts the opposite $2x>2w$, which cannot be true also.
Reaction Set E |  Reaction Set F |  Reaction Set G |  Reaction Set H
(invalid) |  (invalid) |  valid |  valid
1\. $x+z\rightarrow y+w$ |  1\. $2z\rightarrow y+w$ |  1\. $2w\rightarrow 2y$ |  1\. $2w\rightarrow y$
2\. $x+y\rightarrow z$ |  2\. $x+z\rightarrow y$ |  2\. $x\rightarrow w$ |  2\. $y+w\rightarrow 2x$
3\. $x+w\rightarrow 2y$ |  3\. $z+w\rightarrow x$ |  3\. $2z\rightarrow x+y$ |  3\. $x+w\rightarrow y+z$
4\. $x\rightarrow 2z$ |  4\. $x+z\rightarrow y+w$ |  4\. $x+y\rightarrow z+w$ |  4\. $2z\rightarrow w$
valid atom conservation solutions?
NO |  NO |  YES |  YES
$w=0$, $x=0$, $y=0$, $z=0$ |  $w=0,x=z,y=2z$ |  $w=z$, $x=z$, $y=z$ |  $w=2z$, $x=3z$, $y=4z$
valid free energy solutions?
YES |  YES |  YES |  YES
Example: |  Example: |  Example: |  Example:
$w=-305$, $x=-64$, $y=-212$, $z=-375$ kJ/mol |  $w=0$, $x=-237$, $y=-433$, $z=-105$ kJ/mol |  $w=-374$, $x=-41$, $y=-435$, $z=-189$ kJ/mol |  $w=-72$, $x=-206$, $y=-306$, $z=-20$ kJ/mol
1\. $(-64+-375)>(-212+-305)$ |  1\. $2(-105)>(-433+0)$ |  1\. $2(-374)>2(-435)$ |  1\. $2(-72)>-306$
2\. $(-64+-212)>-375$ |  2\. $(-237+-105)>-433$ |  2\. $-41>-374$ |  2\. $(-306+-72)>2(-206)$
3\. $(-64+-305)>2(-212)$ |  3\. $(-105+0)>-237$ |  3\. $2(-189)>(-41+-435)$ |  3\. $(-206+-72)>(-306+-20)$
4\. $-64>2(-375)$ |  4\. $(-237+-105)>(-433+0)$ |  4\. $(-41+-435)>(-189+-374)$ |  4\. $2(-20)>-72$

Table C.1: Examples of Valid and Invalid Abstract Reaction Sets. Each set has
4 species and 4 elementary reactions that are restricted, at most, to be bi-
molecular. Solutions to validity tests 1 and 2 computed with Wolfram Alpha.

### Validity Tests are Not Additive

In general, the validity tests proposed above are not additive. This means
that if two sets of elementary reactions - both involving some of the same
species - each pass both tests individually, then there is no guarantee that
the larger chemical system formed by bringing them together will also pass
both tests.

For example, Reaction Set A in Table C.1 can be separated into two elementary
reaction sets: one composed of reactions (1),(3) and (4), and the other
composed of reaction (2). Individually, each of these sets passes validity
tests 1 and 2, but brought together, the four reaction system fails both
tests.

### Reaction Sets with Non-Elementary Steps

Abstract reaction sets found in the literature often contain non-elementary
reactions. These are reactions which actually proceed as two or more
elementary steps but are written for convenience, or for lack of knowledge, as
a single ‘black box’ reaction representing the overall chemical change222Non-
elementary reactions often have kinetics different from the mass action
kinetics of their overall chemical equation (e.g. see ?, ?). Reactions
catalysed by enzymes are often written in one step e.g. $X+E\rightarrow Y+E$,
but actually considered to proceed in two elementary steps involving formation
of an enzyme-substrate complex. For this reason, catalysed reactions usually
have their rate determined by the Michaelis-Menten approximation (see ?, ?),
rather than by mass action kinetics.. In order to properly apply the two
validity measures above, reaction sets with non-elementary steps need to have
all such steps expanded to their elementary reactions. Then, the measures
should be applied to the entire set of elementary reactions.

Nevertheless, having said the above, abstract reaction systems such as the
oscillating Lotka-Volterra model (?, ?), or minimal bistable systems like
Schlogl’s (?) or Wilhelm’s (?) model, often have their kinetics specified as
mass action kinetics, despite containing steps which are questionably
elementary. This implies that these reactions are treated as if elementary. If
the reactions are assumed to be elementary on these grounds, then the validity
tests can be applied, and the three examples just cited each pass both tests.
The Oregonator (the version presented in ?, ?) also passes both tests.

### Reservoir Analysis of Reaction Networks Limits Stoichiometric
Possibilities

Perhaps for mathematical simplicity, chemical reactions are traditionally
analysed in ‘reservoir conditions’. A set of species is designated as nutrient
species, a set of species is designated as waste species, and these species
sets are given constant reservoir concentrations.333The additive free energy
of the nutrient species is greater than the additive free energy of the waste
species, although individual waste species can have higher free energy than
individual nutrient species. The dynamic behaviour of the remaining
intermediate species is recorded: these are species that are both produced and
consumed in reactions.

However, other reaction sets not suitable for reservoir conditions, but valid
nevertheless, can and do exist. Reaction sets G and H in Table C.1, for
example, pass validity tests 1 and 2 but are not suitable for reservoir
conditions because each one of the species $w$, $x$, $y$ and $z$ is both
consumed and produced. Here, designating a species as a constant concentration
nutrient has the effect of generally breaking up the reaction set into trivial
parts. Instead, the behaviour of reaction sets like G and H can be
investigated by also modelling an explicit reactor context in which the
reactions take place. Rather than the chemical reaction being sustained by
nutrient and waste species at fixed concentrations, it is instead sustained by
chemical flows into and out of the reactor. It follows that a greater
diversity of reaction mechanisms can be modelled in explicit reactor contexts
(including in compartments), than in reservoir conditions.

##  Appendix D Published Manuscripts

Manuscripts published as part of this thesis are linked to in this Appendix
for reference. All manuscripts were published as open access.

### Journal Papers

  1. 1)

Shirt-Ediss, B., Ruiz-Mirazo, K., Mavelli, F. and Solé, R.V. (2014). Modelling
Lipid Competition Dynamics in Heterogeneous Protocell Populations. _Scientific
Reports_ , 4, 5675. doi: 10.1038/srep05675.

14 pages of supplementary material is available online (doi:
10.13140/2.1.2170.1448).

  2. 2)

Shirt-Ediss, B., Solé, R.V. and Ruiz-Mirazo, K. (2015). Emergent Chemical
Behavior in Variable-Volume Protocells. _Life (Basel)_ , 5, 181-211. doi:
10.3390/life5010181.

16 pages of supplementary material is available online (doi:
10.13140/2.1.4242.7206).

### Conference Paper

  1. 1)

Shirt-Ediss, B., Solé, R. and Ruiz-Mirazo, K. (2013). Steady state analysis of
a vesicle bioreactor with mechanosensitive channels. In P. Liò, O. Miglino, G.
Nicosia, S. Nolfi, and M. Pavone (Eds.), _Advances in Artificial Life, ECAL
2013: Proceedings of the Twelfth European Conference on the Synthesis and
Simulation of Living Systems._ 1162-1169, MIT Press. doi:
10.7551/978-0-262-31709-2-ch178.

## Copyright Permissions for Reproduced Figures

Box 1, Figure (a). Reproduced from Pohorille, A. and Deamer, D., Research in
Microbiology, 160(7), 449-456, 2009 (?, ?). Copyright ©2009 Elsevier Masson
SAS. All rights reserved.

Figure 1.2(b). Reproduced from Ichihashi et al., Nature Communications, 4,
2494, 2013 (?, ?). Reprinted by permission from Macmillan Publishers Ltd:
Nature Communications, Copyright 2013.

Figure 1.2(c). Reproduced from Kurihara et al., Nature Chemistry, 3(10),
775-781, 2011 (?, ?). Reprinted by permission from Macmillan Publishers Ltd:
Nature Chemistry, Copyright 2011.

Figure 1.2(d). Reproduced from Noireaux, V. and Libchaber, A., Proceedings of
the National Academy of Sciences of the United States of America, 101(51),
17669-17674, 2004 (?, ?). Copyright 2004 National Academy of Sciences, USA.

Figure 1.2(e). From Krishna Kumar et al., Angewandte Chemie, 50(40),
9343-9347, 2011 (?, ?). Copyright ©2011 by John Wiley Sons, Inc. Reprinted by
permission of John Wiley & Sons, Inc.

Figure 1.2(f). Copyright ©Hardy et al., Proceedings of the National Academy of
Sciences of the United States of America, 112(27) 8187-8192, 2015 (?, ?). (The
National Academy of Sciences of the United States of America, the publisher of
this article, does not require permission to be obtained for reproduction in
non-commercial and educational use.)

Figure 1.2(g). From Luigi Luisi et al., ChemBioChem, 11(14), 1989-1992, 2010
(?, ?). Copyright ©2010 by John Wiley Sons, Inc. Reprinted by permission of
John Wiley & Sons, Inc.

Figure 1.2(h). Reprinted figure with permission from Fallah-Araghi et al.,
Physical Review Letters, 112, 028301, 2014 (?, ?). Copyright 2014 by the
American Physical Society.

Figure 1.2(i). Copyright ©Budin, I. and Szostak, J., Proceedings of the
National Academy of Sciences of the United States of America, 108(13)
5249-5254, 2011 (?, ?). (The National Academy of Sciences of the United States
of America, the publisher of this article, does not require permission to be
obtained for reproduction in non-commercial and educational use.)

Figure 1.2(k). Reprinted (adapted) with permission from Takakura et al.,
Journal of the American Chemical Society, 125(27), 8134-8140, 2003 (?, ?).
Copyright (2003) American Chemical Society.

Figure 1.2(l). Copyright ©Zhu et al., Proceedings of the National Academy of
Sciences of the United States of America, 109(25), 9828-9832, 2012 (?, ?).
(The National Academy of Sciences of the United States of America, the
publisher of this article, does not require permission to be obtained for
reproduction in non-commercial and educational use.)

Figure 1.2(m). Copyright ©Terasawa et al., Proceedings of the National Academy
of Sciences of the United States of America, 109(16), 5942-5947, 2012 (?, ?).
(The National Academy of Sciences of the United States of America, the
publisher of this article, does not require permission to be obtained for
reproduction in non-commercial and educational use.)

Figure 1.2(o). Reprinted with permission from Szymanski et al., The Journal of
Physical Chemistry C, 117, 13080-13086, 2013 (?, ?). Copyright (2013) American
Chemical Society.

Figure 1.2(p). Reprinted with permission from Maselko, J. and Strizhak, P.,
The Journal of Physical Chemistry B, 108(16), 4937-4939, 2004 (?, ?).
Copyright (2004) American Chemical Society.

Figure 2.6. Artwork Copyright © and TM Rube Goldberg Inc. All Rights Reserved.
RUBE GOLDBERG ®  is a registered trademark of Rube Goldberg Inc. All materials
used with permission. rubegoldberg.com.

Figure 2.7. Reprinted from Tibor Ganti, Journal of Theoretical Biology,
187(4), 583-593, 1997 (?, ?). Copyright 1997, with permission from Elsevier.

Figure 3.1(b). Reproduced from McMullin, B. and Varela, F.J., SFI Working
Paper 97-02-012 (?, ?) with permission from Barry McMullin.

Figure 3.1(f). Reprinted from Ono, N., BioSystems, 81(3), 223-233, 2005 (?,
?). Copyright 2005, with permission from Elsevier.

Figure 3.3(a). Reprinted from Ruiz-Mirazo, K. and Mavelli, F., BioSystems,
91(2), 374-387, 2008 (?, ?), Copyright 2008, with permission from Elsevier.

Figure 3.3(b). Reprinted from Wilson et al., Origins of Life and Evolution of
the Biosphere, 44(4), 357-361, 2015 (?, ?), with permission from Springer.

Figure 3.3(c). Reprinted from Ruiz-Mirazo et al., Software Tools and
Algorithms for Biological Systems, Chapter 70, 689-696, 2011 (?, ?), with
permission from Springer.

Figure 4.1(c). Reprinted from Dawson, K. A., Pure and Applied Chemistry,
64(11), 1589-1602, 1992 (?, ?), with permission from Springer.

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