Oral presentation Delving Deeper into Homeostatic Dynamics of Reaction Diffusion Systems with a General Fluid Dynamics and Artificial Chemistry Model Stuart Bartlett1 1Earth Life Science Institute, Tokyo Institute of Technology, Tokyo, 152-8550 Japan [email protected] Abstract re-construct the key chapters in the history of abiogenesis Downloaded from http://direct.mit.edu/isal/proceedings-pdf/ecal2017/29/52/1904544/isal_a_013.pdf by guest on 25 September 2021 on this planet and perhaps others. In this paper I present a general modelling framework for cou- Much of my past work has ventured in this direction, pled fluid dynamics and chemistry problems, and apply it to the simulation of a series of complex, homeostatic reaction and in previous papers I have illustrated the emergence of diffusion systems. The model can incorporate any number of competition (Bartlett, 2014; Bartlett and Bullock, 2015) and chemical species and reactions. Those chemical species dif- homeostasis (Bartlett and Bullock, 2016) within systems of fuse, react and are advected by fluid flows. I illustrate some reaction diffusion (RD) structures. These patterns are spatial characteristic results from the modelling of the Gray Scott instabilities, akin to solitons, manifested as distinct shapes reaction diffusion system with thermally resolved reactions. Extending my previous work on ecological dynamics of non- in the concentration profiles of chemical species in a 2- living structures, I demonstrate that thermal homeostasis of dimensional domain. Their existence serves to channel free reaction diffusion spots can occur in systems without the use energy between two imaginary reservoirs at fixed chemical of the porous wall boundary condition that has traditionally potential. The original model RD systems were introduced been used for the Gray Scott system. I present an initial anal- by Gray and Scott (1985, 1994), though we can of course ysis of the parameter space of this system as well as detailing the mechanism behind the thermal homeostasis. trace the RD concept back to Turing (1952). They represented a seminal new class of non-linear chem- ical system, which inspired a wave of theoretical and ex- Introduction perimental work (see e.g., Awazu and Kaneko, 2004; Lee Alongside the dissipation of free energy and autocatalytic et al., 1993, 1994; Lee and Swinney, 1995; Mahara et al., growth, a fundamental property of the living state is the 2008; Pearson, 1993; Virgo, 2011). One of the most striking ability to regulate a subset of local variables to within win- features of these systems, under certain parameter regimes, dows of specific set points. This ability to self-regulate, en- is the emergence of self-replicating spot patterns, strongly capsulated by the concept of homeostasis, distinguishes life reminiscent of the multiplication of single-celled organisms from many non-living dissipative structures such as fire. Fire (often when I play animations of RD systems to the unini- dissipates free energy and grows exponentially given am- tiated, their first question is whether they are looking at a ple supply of fuel and oxidant. What it is not able to do is video of bacteria). It was these systems that I experimented perform any kind of feedback with its external environment with using simulations in previous work. I found that mul- such that its behaviour does not remove the conditions for tiple sets of spot species could spontaneously regulate their its own existence. Once a forest is burned, the fire stops. An local temperature, using a primitive form of reign control adaptive fire would sense any external decline in fuel supply (Bartlett and Bullock, 2016), similar to the manner in which and eventually tune its burning rate to match the rate of tree populations of Daisies regulate temperature in the Daisy- regrowth. That would allow it to persist far longer than its world model (Watson and Lovelock, 2011; Wood et al., equivalent, ravenous cousin that consumes with disregard. 2008). One of the key objectives of Artificial Life is to estab- This and similar works allow us to consider general, prim- lish general conditions under which the living state emerges. itive means by which physical and chemical dynamics can Hence we continually seek simple model systems that ex- lead to self-regulation. In order to explore such phenomena emplify properties associated with life, despite the given further, it is necessary to have modelling techniques that are system being non-biological. Once all such necessary self- physically realistic but also permit a large number of degrees organisation processes are charted and understood in the ab- of freedom. This was my primary motivation for developing stract, we should then be able to map them to more realistic a reactive, thermal lattice boltzmann model (RTLBM) capa- and geologically plausible conditions, and thus attempt to ble of handling a vast range of chemistries, while also retain- Carole Knibbe et al, eds., Proceedings of the ECAL 2017, Lyon, France, 4-8 September 2017, 52 (Cambridge, MA: The MIT Press, ©2017 Massachusetts Institute of Technology). This work is licensed to the public under a Creative Commons Attribution - NonCommercial - NoDerivatives 4.0 license (international): http://creativecommons.org/licenses/by-nc-nd/4.0/ Oral presentation ing a faithful coupling to the hydrodynamics of the system. the non-linear reaction responsible for pattern formation is In this work I will present this very general fluid dynamics present: and artificial chemistry model, and use it to extend my previ- ous work on homeostasis in the Gray Scott system. Regulat- Ai + 2Bi )* 3Bi i 2 f1; :::; Ng: (2) ing a single variable using two populations of individuated Note that there is the possibility to include not just one or structures was a significant result. However the thermody- two, but N such ‘spot species’ and that the reaction is re- namic gradients placed across those systems were somewhat versible. I retain the reversibility for the sake of generality. constrained. They used the standard, homogeneous porous For the above reaction, the reverse direction is forced to be wall (PW) boundary condition (BC). In order to have full negligibly slow by imposing a very low frequency factor. control over the geometry and spatial orientation of the sys- Allowing a stronger reverse reaction would warrant its own tem’s gradients, it was necessary to find an alternative set dedicated investigation, which can be taken up in the future. of BCs, such that supply and removal of chemical species Instead of using the standard PWBCs, in this work I will could be arbitrarily adjusted. Thus I will present a method use an extra two sets of chemical species (two extra chemi- for achieving this wherein two extra sets of chemical species Downloaded from http://direct.mit.edu/isal/proceedings-pdf/ecal2017/29/52/1904544/isal_a_013.pdf by guest on 25 September 2021 cal species per spot species), which will produce an equiva- are introduced. The inflow and outflow of these two species lent supply and extraction effect. There are several motiva- defines the chemical potential dissipated by the system, and tions for making this change. Firstly, the PWBCs introduce it can be adjusted in any desirable way. I will demonstrate their own negative feedback into the system since they act the reproduction of my previous results on thermal home- to maintain the concentration of the A species close to 1. It ostasis and go on to present a first parameter space mapping i would be useful to be able to isolate and remove this neg- of this system’s behaviour. ative feedback effect to assess whether other negative feed- In the following section I will outline the equations of mo- backs are sufficient to stabilise a given set of patterns. Sec- tion that are solved by my numerical method. Precise mod- ondly, it allows all the dynamics of the system to fall within elling details can be found elsewhere (Bartlett, 2014; Bartlett the classes of reaction, diffusion, and fluxes at outer bound- and Bullock, 2015, 2016). I then present results of my simu- aries. Inward/outward fluxes at every point in the system of lations including steady state configurations, population dy- different chemical species would be difficult to implement namics and how resilience varies with two key parameters experimentally, and don’t extend naturally to 3D systems (if of the system. I discuss the results and suggest directions we were envisaging that system being re-created in the lab). for future work in the final two sections. Thirdly, it allows us to clearly visualise flows of material Model System across the system in an easily recognisable way. It will also allow us to experiment with, for example, sys- The simulations in this work make use of the RTLBM (for tems where there are no external material fluxes but only full details, please see Bartlett, 2014; Bartlett and Bullock, energy fluxes (e.g. heat flow). The material within the sys- 2015, 2016), that was developed as an extension of simi- tem would be allowed to pass through cycles of reactions lar models (Ayodele et al., 2011, 2013; Frouzakis, 2011; He (reversibility of the reactions would be crucial), probably et al., 1998; Peng et al., 2003; Shan, 1997). In this section, I passing through exothermal transitions in low temperature will simply outline the governing equations for all the vari- regions and endothermic transitions in higher temperature ables simulated. regions. We can then experiment with the effect that com- In all cases, the system in question is a rectangular two plex pattern formation has on the heat transport abilities of a dimensional fluid with a 4:1 aspect ratio bounded by solid system (interactions with convective flow structures). walls enforcing the no-slip velocity BC. The flow of the fluid The more general BCs presented in this work allow an obeys the standard, incompressible Navier-Stokes equation: arbitrary range of spatially directed gradients.