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This is an open set of experimental notes, containing links to analysis tools, data, and software. In addition it is supplementary material for a paper submitted to the autonomy workshop at ALife X. People are encouraged to contribute if they are interested. All criticism is very warmly welcomed. Anyone who wishes to collaborate on papers based on material in these pages is very welcome to contact me and contribute to these pages and work on papers together. These pages contain very new research. It is my hope that sharing these ideas will lead to increased productivity due to contributions by people with complementary skills.

A Toy Thermodynamic Generative Chemistry

Consider the following artificial chemistry that intends to help us understand the minimal conditions required for chemical systems to become increasingly complicated. The hypothesis that I am testing applies to a chemical network that is closed to matter but open to energy, i.e. capable of recieving high quality energy (photons), and extruding heat. I suggest that increasingly complex chemical organizations will arise if it is possible for high energy chemicals to become transformed in some way such that they catalyse the synthetic reactions that produced them. This requires that their reactivity be curtailed, i.e. that there is some means by which kinetic control can be exerted on their reactivity. What sorts of properties must chemical reaction systems have if they are to maximize the rate of heat production, i.e. to reach increasingly ordered configurations that are capable of improved heat conduction?

Rules for Constructing Reactions

Atoms consist of letter, a,b,c,d,.etc... and molecules consist of strings of such letters. The order of the string is important.

Rule1: Generate initial endothermic (anabolic) reactions: Starting from the primitive letters, generate 50 chemical reactions of the form {X + p -> Y} , {X + Z + p -> Y}, {X + p -> Z + Y} and {X + Z + p -> Y + Q}, with the only constraint being that there is conservation of atoms. p represents a high quality photon, or a quantum of useful energy. This set of reactions is the set that is capable initially of absorbing external energy, p, therefore, by definition the energy of the products of these reactions will be greater than the energy of the reactants. To make this explicit, we assume that p has energy unit 10. When a reaction is generated, some of the species in the reaction may already have been assigned free energies previously. If this is the case, then the free energies assigned to other species in the reaction is constrained by this. If no suitable free energy combination is possible that ensures that the same species has the same free energy in all reactions, then the reaction is disregarded. Thus, the generative algorithm ensures that both energy and mass is conserved in the initial set of endothermic reactions. For example a sequence of generative events is shown below.

1. [ a + p ---> a ] //This reaction is rejected since there are no values of a that can ensure energy balance, since we cannot solve the simultaneous equations Ea + Ep = 10, Ea = 10, and Ep = 10, where Ex is the free energy of species x.

2. [a + b + p ---> ba] //No free energies (E) have been assigned to any species. Therefore, we assign a random value to the reactants, a an b, e.g. 3 and 4, meaning that ba MUST have a free energy of 17. The species list so far is as follows , [a-3, b-4, ba-17] and the reaction list is [a+b+p->ba].

3. [a + ba + p -> aba] //Free energies are already assigned for a and ba, of 3 and 17 respectively, therefore, aba must have a free energy of 3 + 17 + 10 = 30. Species list = [a-3, b-4, ba-17,aba-30], reaction list = [a+b+p->ba, a + ba + p -> aba].

4. [a + a + p --> aa] //Ea = 3 therefore Eaa = 16. Species list = [a-3, b-4, ba-17,aba-30, aa-16], Reaction list = [a+b+p->ba, a + ba + p -> aba, a + a +p -> aa].

5. [ba + p --> a + b] // ba, a, and b have all been assigned free energies and it is impossible to tally these with this reaction, since ba = 17, a = 3 and b = 4, and so this reaction is rejected.

6. [aba + p ---> aa + b] // Eaba = 30, Eaa = 16 and Eb = 4, therefore this reaction is rejected.

6. [aba + aa ---> aaaa + b] //Eaba = 30, Eaa = 16, Eb = 3, therefore, Eaaaa = 30+16+10-3 = 53.

This process continues, until 50 such reactions have been generated. It may well be necessary to scale these free energy values so that they lie within a smaller range so that simulation is possible. Neverthaless, at the end of this process a particular chemical network is produced. So far you notice that no exothermic reactions capable of dissipating energy have been generated. It is time to generate these.

Rule 2: Generate initial exothermic (catabolic) reactions: Take a chemical species that was generated previously and generate a dissipative reaction, e.g. aaaa ----> aaa + a.

Reactions are of several types, precisely as before, but without the inclusion of p. Instead, the constraint exists that products must have lower energy than the reactants.

i. {X -> Y} //Dissipative rearrangements.

ii. {X + Z -> Y} // Dissipative Ligations.

iii. {X -> Z + Y} //Dissipative Cleavages.

iv. {X + Z -> Y + Q}, //Dissipative bimolecular rearrangements.

The reactions are generated by taking existing molecules and generating new molecule types, checking whether these have already been created, and constraining the reactions if this is the case. Again, we try to balance the energies and allow the reaction only if this balancing is possible, but forbid the reaction if the balancing is not possible. In the above reaction, Eaaaa = 53, Eaaa is undefined and Ea = 3, therefore Ea < 50. If Ea is 50, then the free energy on both sides of the reaction is the same and the reaction is at equilibrium. Therefore some random number is generated below 50, and assigned to Eaaa. The new species set is thus for example, Species list = [a-3, b-4, ba-17,aba-30, aa-16, aaaa-53, aaa-25], Reaction list = [a+b+p->ba, a + ba + p -> aba, a + a +p -> aa, aba + aa -> aaaa, aaaa --->aaa + a]. aaa has been assigned the energy 25. This means that this reaction is capable of dissipating 25 units of energy for one reaction event, or for unit flow. It is possible to allow a back reaction, in which case the equilibrium position of the reaction will depend on the temperature of the system. We include such reactions typically, so aaaa <---->aaa + a. Later we describe how rates for each reaction are calculated 50 such reactions are generated. Of-course, it will be the case that some products are also those species that are capable of absorbing energy and these will be re-cycled.

An important kind of reaction that is generated at this stage is a coupling reaction, or energy exchange reaction, e.g.

[ aaaa + b ---> aaba + a ] //Eaaaa = 53, Eb = 4, aaba is undefined and a = 3, therefore Eaaba < (53+4-3 = 54). A new species has been created that can take part in further reactions,depending on the value of that species free energy. One possibility is that exothermic reactions loose a large amount of free energy. This results in less reactive speces, but a large generation of heat. Another possibility is that species can reactants have high energy.This results in a lower dissipation of heat, but a larger number of reactants that are themselves capable of taking part in reactions that dissipate heat.

Rule 3: Generate reaction rates: For simplicity we choose to assume that all endothermic rates = 1, as are the exothermic forward reaction rates. The backward exothermic reactions are given rates that result in the correct equailibrium ratio K expected from the difference in free energies between products and reactants. Since K = e^{-dG/RT} where dG is the difference in free energy i.e. Free energy of products minus free energy of reactants. Where dG is negative, it means that free energy is lost in the reaction, and so the reaction is tipped towards the products, i.e. K is high. This means that the backward catabolic reactions will have to have rates < 1 in order to produce the correct K value for those reactions. A more realistic distribution of kinetics will be considered in further experiments.

Rule 4 : Simulate Chemical Reactor: The system of reactions is coded up into a stoichiometric matrix and simulated using standard ODEs with Eular integration until some steady state is obtained. The heat dissipation at this steady state is calculated as the quantity of flux through exothermic reactions multiplied by the difference in free energy of those reactions. Any particular chemical network will have a certain capacity to maintain a certain heat dissipation. In order to make visualization easier, we can simulate the chemical network evolving on a surface. This will also allow the investigation of diffusion limiation, protocell formation, etc... e.g. some chemical species can be permitted to limit te diffusion of others, and provided with physical properties such as amphipaticity.

Rule 5: Stochastically Generate Novel Reactions: What sorts of generative chemical process may result in increasingly complex and stable chemical organizations that are able to increase the rate of heat dissipation in the system, and which sorts of generative chemical process will not permit this? How likely are such generative processes in real chemistry? One of the problems with the binary string chemistry described in my DPhil was that high energy particles were not able to obtain kinetic stability since they could react with any other chemical with ease, and reach equilibrium.

One method to obtain novel particles is to allow a particle to undergo a novel reaction to produce a new particle. There are several ways in which particles can be chosen that will undergo novel reactions, i). randomly, ii). as a function of the particles free energy, iii). as a function of particle composition. Initially we consider the random case. Secondly, there are several ways in which the chosen particle could react, the most obvious distinction is between a new reaction that is endothermic, i.e. absorbs energy (p) , and a reaction that is exothermic, i.e. gives off energy. The probability of the two novel reaction types can be altered. The likelihood of novel endothermic reactions capable of utilizing more p can be completely independent of the reactant type, or we may specify that certain species configurations are capable of utilizing certain energy sources, e.g. a palindromic sequence may be assigned p absorbing capacity for example. There may be a new reaction between two already existing particles, in which case tests will have to be carried out to see if the reaction is thermodynamically reasonable.

Alternatively, a new particle may have similarity in its reaction profile with the particle from which it came, i.e. if abb -> abbb, we might allow abbb to have all the same kinds of reactions as abb, but with small differences, e.g. the possibility of loosing or gaining a reaction. This would simulate the capacity for small alterations in the active sites or reactive groups of organic molecules by for example methylation. The capacity for such small alterations may be essential in the self-organisation of chemical systems.

When the system settles to equilibrium, a new reaction is introduced. This reaction may drive some species to extinction. If the concentration of species drops below 0.001, and if that concentration remains below that value for some time, then that species and any reactions generating that species are removed. This resembles community iteration models, except here we are applying a similar methodolody to chemical reaction systems! The validity of the movement of the simulated window through chemical space depends on the way it is done.

Results and Programs in Progress

We begin with the usual framework code used for running simulations, downloadable here. The usual world is generated, and reaction and species objects are created. Reactions are generated. Downloadable here is a version that generates all the endothermic light absorbing reactions. Downloadable here is a version that generates all the endothermic light absorbing reactions and the exothermic heat generating reactions. Downloadable here is a version that generates the reaction rates for the previously created reactions. Downloadable here is a version that actually runs the reactions, without introducing any new reactions. The chemical concentrations are dumped to a file chemicals.data

An example of species concentrations generated using the simulator with a randomly initialized set of species is shown here. Only the first set of species 'a', 'b', 'c', 'd', 'e', start off with non-zero 'concentrations' of 100. Concentrations are on the y-axis, and time-steps on the x-axis.

The system settles into an equilibrium of-course. Photons are supplied at a constant concentration 0.1. We can measure the extent of heat dissipated by the system, and the extent of free energy within the system. The two graphs below show this for a randomly initialized system consisting of 20 reactions. 10 light absorbing, and 10 heat producing. The top graph shows the chemical concentrations, and in red the total free energy of the system over time. The bottom graph shows in a log plot over time where yellow is the rate of light energy absorbtion by the system, and blue the net heat production by the system. You can see that both these values decrease over time.

It is evidant that this randomly generated chemical network is not very good at capturing energy or at dissipating heat. Compared to the starting chemical energy of the system the energy captured from light does not increase very much. What sort of process is required for the network to self-organize in order to increase the extent of heat dissipation, and hence energy absorption? Several possibilities are imaginable,

i. Resynthesise the molecules that are capable of absorbing energy.

ii. Generate new molecules that are capable of absorbing energy and re-synthesise those.

What reasonable process can allow the above changes to occur? Once the network has reached a quasi-steady state, i.e. after a certain period of time after which the concentrations are not changing very much, we allow the creation of new reactions. There are several ways in which a new reaction could be created.

Regimes for Generating New Reactions

i. A random species is chosen from those that exist, and a new reaction involving that species is carried out. If the reaction is a bimolecular reaction then another species must also be chosen at random. The reaction could either be an exothermic reaction, or it could be an endothermic one that absorbs light energy.

ii. Although a random species is chosen to undergo a reaction, the reaction may not be random. e.g. during a re-arrangement reaction for example, it may be the case that the resulting product is capable of undergoing similar reactions as the original molecule, but with some structued changes, e.g. let the new species react with all the same molecules that the old species reacts with but with either a few missing reactions, or with a few additional reactions.

iii. A species may be chosen in proportion to its concentration, to undergo a random new reaction.

iv. A species may be chosen in proportion to its free energy to undergo a new random reaction. This has the justification that in general species with greater free energies are more likely to undergo reactions with other species. However, this generative rule may result in high energy species being kinetically unstable, i.e. reacting very non-specifically with other species. It may be more realistic to allow high energy species to produce varients of themselves with slightly altered reaction profiles, i.e. plus or minus a single reaction and slight variation in reaction rates.

v. A species my by chosen in inverse proportion to its free energy, to undergo a new random reaction.

vi. A new reaction may be constrained to have a product that is a species which already exists. This forces cyclic reaction systems to arise.

vii. A new reaction may be constrained to have a product that is a species that itself can already absorb light, i.e. all new reactions are forced to be re-cycling reactions.

viii. So far there has been no notion of new species being able to increase the rate of light absorbing reactions, in fact it is only through increasing the extent of re-cycling that species are capable of increasing the rate of photon absorption. Here we introduce explicit catalysis terms and allow species to increase the rate of an existing light-absorbing reaction with a certain probability.

This paper will simply examine the consequences of these different reaction generation regimes for the resting free energy of the system and the system's capability of heat dissipation over time.

Regime 1: Random introduction of new reactions (without reaction removal).

The system was initialized with only 'a' type atoms at the begining, 100 endothermic, and 100 exothermic reactions of all molecularities were generated. Species were chosen randomly to undergo a new reaction which could be of any one of the 8 possible types, with equal probability. This tended to produce very large species due to the abundance of bimolecular ligation reactions.

It is important to visualize the features of the chemical reaction network, e.g.

o. Check that mass conservation and energy conservation are occuring. Demonstration of mass conservation involves multiplying the concentration of a species by the length of that species, for all species, and checking that the sum does not change. Demonstration of energy conservation involves i). removing external energy input and checking that the heat produced + internal energy = constant, ii) checking that input energy + internal energy - output energy = constant.

The above graph shows a run where 100 light absorbing and 100 heat producing reactions are randomly generated. No light is supplied to the reactor, i.e. Pconc = 0, and no new reactions are generated. You see that the concentrations settle down quickly. The total mass of the system does not change (blue line), and the rate of heat production reaches a steady state. Why is there always a very small rate of heat production in the system even though no light is being absorbed? There is an initial phase at the start of the run where the rate of heat production increases to a stable value. This is presumably because the system is not started off at equilibrium, since the initial concentrations are only non-zero for the seed set, in this case, 'a', is the only seed molecule for the entire system. Heat is generated as the species distribute themselves over the reaction network. Note that since there is no light, none of the irreversible endothermic reactions have any flux through them, therefore, matter only distributes itself through the exothermic reactions.

Next we test the influence of allowing light entry without novel reaction generation.

Above, light is constantly present at Pconc = 0.01, with energy, Ep = 0.1. Several species are produced, but over time the rate of light absoption and heat production and internally stored chemical energy decreases. Another three random initializations with the same parameters is shown below for comparison, but with timestep = 0.001 instead of 0.01 as above.

The tendency for light absorption to decrease, and heat production to decrease is present in all cases of randomly initialized networks with fixed light input. As a further control, we see the consequence of not allowing exothermic reactions to be reversible, with all else as above.

In this case the heat out - heat in = heat out, since there is no heat in by reversible exothermic reactions. Apart from that all else looks the same. We can conclude from the above graphs the mass and energy conservation occurs in an abstract sense at least.

i. The distribution of heats produced by exothermic reactions with different generative regimes.

Above is shown the distribution of heats lost in exothermic reactions in three different generative methods. On the top left, and in black dots you see the distribution when 1000 endothermic and then 1000 exothermic reactions are generated consecutively at the initialization of a run. In red is the same plot but with only 100 of each reaction type generated at the begining of the run. In green you see the distribution where the system starts with only one endothermic and one exothermic reaction and reactions are generated randomly. In the later case you notice that the range of heat dissipated in exothermic reactions is much lower compared to the generative algorithm which allows the chaining of many endothermic steps before any exothermic reactions are allowed (black dots). Typically in all further experiments the green generative model is used, starting with only one reaction and allowing random generation of endothermic and exothermic reactions.

ii. The distribution of species free energies as a function of species properties,e.g. length. Free energy tends to increase with string length. Species free energies are much less when there is random generation of exothermic and endothermic reactions from a network starting with one reaction, compared to when 500 endothermic reactions are generated first, followed by 500 exothermic reactions. This is to be expected since the later forces chaining of light absorbing reactions.

ii. The distribution of reaction rates below, shows again that there is much lower variance and mean reaction rates when reactions are generated sequentially, rather than as a batch at the start of the run. Reactions with higher and higher rates get generated later in the run, since typically, species with higher free energies appear later in the run.

All the above results are for the non-cyclic generation regime, i.e. where there is no constraint on at least one of the products already existing. The distributions are expected to be quite different for the case where one of the products must already exist for the reaction to be legitimate.

iii. The topology of the network produced. The number of reactions per species is plotted on the following graphs.

iv. The total rate of photon absorption over time. The total rate of photon absorption is plotted on the following graphs.

Q1. Does the rate of energy dissipation by the system increase over time? Does the rate of light absorbed by the system increase over time?

Q2. Does the amount of energy stored as chemical energy in the system increase over time?

Q3. What is the structure of the chemical network that is generated? i.e. how connected is it, cliquiness etc???

 

Regime 2: Random introduction of forced cyclic reactions (without reaction removal). Download this version

The other extreme case is to force all new reactions to produce at least one species that already exists.

Version 2.0. (May 2006)

Several modifications to the above model have been considered, and implemented. These new models have been considerably influenced by community iteration/construction models.

Details of development of this model are here.