Introduction.
The low probability of random formation of a complex autocatalytic cycle (i.e. a self-replicating chemical cycle with many intermediate constituents) was highlighted by G.A.M King in 1982. King randomly generated theoretical recycling chemical networks consisting of bimolecular reactions, and found that the probability of a viable large autocatalytic cycle was extremely low. However, the possibility of more complex stable reaction topologies was not considered, for example, the existence of side-reactions that later drain back into the autocatalytic cycle, thereby establishing an 'autocatalytic system', of coupled autocatalytic cycles. A model of chemical evolution was designed to help explore the environmental and chemical conditions under which increasingly complex autocatalytic systems could arise.
Unlike other artificial chemistry models, the model described here implements 'realistic' conservation of mass and energy. It is possible to simulate either an open, closed or isolated chemical system. A spin-glass method is used to randomly generate the Gibbs free energies of formation for each molecule. From these free energies, the equilibrium ratios of all possible reactions can be calculated. Since the intermediates of reactions are not modeled, the activation energies for reactions is underdetermined, and therefore we randomly generate the actual reaction rates from an exponential distribution, ensuring that the forward and backward rates of reaction conform to the K value of that reaction. The influence of this kinetic distribution, and the influence of changes to the stoichiometric characteristics of the chemical network, upon the probability of forming autocatalytic systems can be examined.
The most difficult part of the work involves analysis of the chemical network behaviour, since the system exists in a well mixed reactor, and the variables of the system consist of a list of chemical concentrations. Methods are developed to automate the detection of autocatalytic systems.
Methods.
Java Implementation.
The class structure is shown here.
Results
Control Experiments.
The reactor was initialized with 100,000 molecules of A and 100,000 molecules of B. An initial transient occurs in which A and B is rapidly converted into a range of product molecules, see figure 1. The maximum length of particles was limited to 7.
Figure 1. (Top) y-axis shows particle number, x-axsis shows time. There is an initial transient exponential decay of A and B. (Middle) Eventually an apparent equilibrium is reached with stochastic fluctuations. There are a few molecule types with large equilibrium particle numbers, but most molecules exist only at small particle number. (Bottom) The y-axis shows particle number, and the x-axis shows particle free energy. At equilibrium the particle number is related to the free energy of that particle. As the free energy of formation decreases, the particles tend to persist at higher equilibrium concentrations. Note the existence of some particle types with high free energies of formation and yet relatively high equilibrium particle numbers.
In order to observe the behaviour of particles at lower concentration, a log plot is shown in figure 2.
Figure 2. Log of particle number at equailibrium.
All the particle types that are ever going to exist, come into existence very early on in the simulation. This can be seen in Figure 3 which shows the first appearence of particles occurs within the first few time units of initialisation.
a)
Figure 3a. The y-axis shows particle type, with strings of increasing length going up. The x-axis shows Log[time]. Most particles are generated within the first few 'seconds', after initialisation. Dots show the presence of that particle type at that time.Particle concentrations are sampled every 100,000 reaction events. There are 253 possible strings (of length up to 7) in this universe. Each type exists at non-zero concentration at equilibrium. This is unlike the real universe where the number of possible molecule types far exceeds the number of actual molecule types. Similar results are observed for maximum string lengths of 9, where all the 1021 possible molecules quickly come into existence (see Figure 3b). The mathematica code for generating this figure from the data files produced by the simulation code can be found here.
The distribution of reactions and their kinetics is shown in figure 4.
1. Effect of altering the minimal kinetic threshold of the simulation, (minRelaventRate).
If the kinetic threshold of the simulation is increased, then the moleculer types come into existence more slowly and posess different equilibrium concentrations, however, the full range of molecule types is produced.
2. Effect of altering the distribution of the rates of reaction.
Bibliography
Anderson, P.W. Suggested Model for Prebiotic Evolution: The Use of Chaos. (1983) Prof. Natl. Acad. Sci. USA 80, 3386-3390 Download pdf.
Benko, G. Flamm, C. Stadler, P.F. A Graph-Based Toy Model of Chemistry. (MCC 2002, Dubrovnik) Download pdf.
Benko, G. Flamm, C. Stadler, P.F. Generic Properties of Chemical Networks: Artificial Chemistry Based on Graph Rewriting. Download pdf.
Benzhaf, W. Self-Organisation in a System of Binary Strings. (1994) Download pdf.
Benzhaf, W. Self-Replicating Sequences of Binary Numbers. The Build-up of Complexity. (1994) Download ps.
Dittrich, P. Ziegler, J. Banzhaf, W. Artificial Chemistries - A Review. (2001) Download pdf.
Farmer, J.D. Kauffman S.A. Packard, N.H. Autocatalytic Replication of Polymers. (1996) Physica 22D, 50-67. Download pdf.
Hordijk, W. and Steel, M. Detecting autocatalytic, self-sustaining sets in chemical reaction systems. Journal of Theoretical Biology (2003, In Press) Download pdf.
King, G.A.M. Recycling, Reproduction, and Life's Origins. (1982) BioSystems, 15, 89-97 Download TIFF.
Krishna, S. and Jain, S. Graph theory and the Evolution of Autocatalytic Networks. Download pdf.
Milo, R.et al. Network Motifs: Simple Building Blocks of Complex networks. (2002) Science, 298, 824-826 iPhoto.
Schilling et al. Metabolic Pathway Analysis: Basic concepts and scientific applications in the post-genomic Era. (1999) Biotechnol. Prog. 15, 296-303. iPhoto.
Schuster, S. Fell, D.A. Dandeker, T. A general definition of metabolic pathways useful for systematic organization and analysis of complex metabolic networks. (2000) Nature Biotechnology, 18, 326-332. iPhoto.
Stein, D.L. and Anderson P.W. A Model for the Origin of Biological Catalysis. (1984) Proc. Natl. Acad. Sci USA 81 , 1751-1753. Download pdf.
Wuensche, A. Basins of attraction in network dynamics. A conceptual framework for biomolecular networks. To appear in modularity in evolution and development. (2004) Download pdf.