Los Alamos - A SUMMARY LA-UR M. W. Olesen DYNAMICAL HIERARCHIES. S. Rasmussen, N. A. Baas, C. L. Barrett,

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1 LA-UR Title: Sumitted to: DYNAMICAL HIERARCHIES - A SUMMARY S Rasmussen, N A Baas, C L Barrett, M W Olesen Artificial Life & Rootics (AROB), Oita, Japan, Feruary, 1996 Los Alamos NATIONAL LABORATORY Los Alarnos National Laoratory, an affirmative aclionkqual under contract W-7405-ENG-36 By acceptance of this articl pulish or reproduce the pulished form of this contriution, requests that the pulisher identify this article as work perf0 fornia for the US Department of Energy ns a nonexclusive, royalty-free license to The Los Alamos National Laoratoty Form No 836 R5 ST OB1

2 I Dynamical Hierarchies - A Summary Steen Rasmussenal Nils A Baasc Christopher L Barrett"t Michael W Olesen" SantaFe Institute OTSA-DO/SA MS-M997 and CNLS MS-B Hyde Park Road Los Alamos National Laoratory Santa Fe, New Mexico, Los Alamos, New Mexico, USA USA Keywords Eigher Order Emergence, Constructive Dynamical Systems, Molecular Self-Assemly, Self-Organization, Cellulur Automata, Lattice Gases, Theory of Simulation Summary This paper summarizes some of the prolems associated with the generation of higher order emergent structures in formal dynamical systems In iological systems, higher order hyperstructures occur oth in an intuitive and a formal sense: monomers, polymers, memranes, organelles, cells, tissues, organs, etc constitute an oservale hierarchy, apparently generated y the underlying iomolecular process However, in models and simulations of these systems, it has turned out to e quite difficult to produce higher order emergent structures from first principles The first prolem is to agree on what a higher order structure is An emergent structure can e defined through an introduction of an oservationalfunction [l]if a property can e oserved in the dynamics, ut not at the level of the fundamental first order interacting structures, we define it to e emergent It is well known that second order structures occur relatively easy in simulation, so the prolem is how to proceed to third and higher order without external interference [2] A third order structure is defined through the interaction of second order structures forming a new oservale not found at the lower levels We can discuss these prolems in the light of a Lattice Gas Automata [4] style discrete field automata, a Lattice Polymer - or Molecular - Automata (LPM, LMA) [8, 51 for molecular self-assemly where we can demonstrate the generation of higher order structures from first principles [2] These formal systems are interesting in their own right, since this level of descrip- =Departmentof Mathematical Sciences University of llondheim, NTH N-7034 Tkondheim Norway tion allows the generation of entropic and enthalpic flowsin a microcanonical, molecular ensemle ringing insight to entropy-driven processes in molecular many-particle systems Structural features of a polar solvent can e generated together with clusterformation of hydrophoic monomers and polymers in a polar environment Polymerization processes, and the self-assemly of micelle-like structures of lipid polymers can also e followed Memrane staility, microtuule assemly/disassemly and cytoskeletal reorganization can also e generated However, the central issue here is dynamics and not molecular selfassemly For a detailed discussion of the physicochemical properties of these self-organizing processes we refer to [5] For an alternative approach to the dynamics of self-organization we refer to [3, 71 The cellular automata we use to descrie the dynamics of these simple molecular self-assemly systems can e formulated as interacting ojects of the form where Si is the ith molecule, zj its position on a 2-D lattice, yj its current state: velocity, kind of molecule, and ond directions; Zj its neighorhood, incoming force particles from neighoring molecules j ; and f i j, the oject-oject interaction rules f i j ( z i ( t ),~ i ( tzi(t)) ), 4 (xi(t + l ), ~ i (+t I)), (2) which change the location and the internal state of current oject depending on the way the oject updates are scheduled (which eg can e random or parallel) For a detailed description of the rules in (1) and (2), their assumptions, and the LMA dynamics, we refer to [8, 5, 21 For an alternative cellular automata approach we refer to [SI If we define monomers to e first order structures, then polymers will constitute second order structures Third order structures will then e given y micelles Polymers carry second order emergent properties such

3 , as elasticity; micelles and memranes carry third order emergent properties such as permeaility - and an inside and an outside Defining a dynamical system as an LMA, it ecomes possile to generate higher order emergent structures from first principles as can e seen in figure 1 It should e noted that this particular process is chosen, ecause of its conceptual clarity and not ecause it models any particular iomolecular process It is clear from the oservations we have made of the dynamics of the discrete field automata systems, that their aility to produce emergent structures is highly dependent on the degree of detail - or fidelity of the ojects in (1) As more and more interactions ( f i j ) and more and more different molecules and molecular states (yi) - are taken into account, the more complex emergent structures the lattice automata systems are ale to produce For example allowing only a simple moleculemolecule interaction without any excluded volume, enales us to define Lattice Gases [4]which can generate a variety of macroscopic fluid dynamics phenomena By defining an excluded volume for the monomers together with inding and scheduling information to each of the molecules, it ecomes possile to generate polymer dynamics These are examples of second order emergent phenomena, as we have mentioned earlier If inding information is present and the initial configuration is a random configuration of polymers with hydrophilic heads and hydrophoic tails the formation of micelle-like aggregates ecomes possile However, in a direct way the polymers can e generated y the dynamics if we allow an interaction which is specific for a polymerization process If ond information together with polymerization interactions as well as hydrophoic/hydrophilic molecules all are defined, it ecomes possile to generate micelle-like polymer aggregates from an initial condition of random monomers as seen in figure 1 Intermediate configurations of the dynamics will then e dominated y the newly polymerized hydrophoic/hydrophilic polymers Thus, it is possile to produce third order structures from first order structures Another way of saying this, is that a more detailed description of the Physics is necessary to allow the formal system to produce higher order structures l Weaker effects also have to e taken into consideration if more complex structures have to e explained - 'We use Physics to denote principles of Nature independent of our description or knowledge of it We use physics to denote our formal understanding and models of these principles = : " - =- = r ',: <,: * H f' 2' e # 0 f: #%:: a,:e 0 *I 8,,& ao 0 e4 l U U f o * *e >--- o'o,o 2 * I HO e m a 4 : ' 0': --A Figure 1: Generation of third level structures in the lattice avtomata system (monomers at time t = 0) + (polymers at time t = 30) 4 (micelle-like aggregates at time t = ) Initially there are approximately the same numer of hydrophoic (lack) and hydrophilic (white) monomers The polymerization occurs such that a hydrophoic monomer (which is the nucleation center) can form a ond to a hydrophilic monomer which in turn can polymerize another hydrophoic monomer etc At time t = 30 there are still a few free hydrophilic monomers not yet polymerized Note the small hydrophoic cluster in the lower right corner of the lattice The formed micelle-like clusters are stale for these initial conditions and parameters

4 Since the very eginning of the study of Artificial Life the following dogma has een dominating: COMPLEX, DYNAMICAL STRUCTURES h SIMPLE RULES AND STATES Is this really true? Could our complex hyperstructures have een created only starting with very simple oject states and interaction rules without any external interaction with the system? We dout it The notion of a critical oject complexity is presumaly important for the generation of dynamical hierarchies up to a given level Whether more explicit internal oject complexity always will e necessary to otain yet higher order emergent structures - or whether there is an oject complexity limit aove which any emergent level can e produced, we do not know We would like to elieve that the latter is true See also [2] By extracting some of the formal principles involved in the formation of a system capale of generating higher order (hyper-) structures in molecular self-assemly systems and relating them to dynamical systems, one of the central issues ecomes the notion of a simulation as a synthetic mathematical method We may define a simulation as a dynamical system that formally is constituted y an ensemle of ojects si = Si(fij,Zi,Ti)r (3) {si@ + 1)) = U{Si(t)}, (4) i = 1,,n, where Si is an oject with internal state si, oject-oject interaction function fii (which typically will have its own state z i as an argument together with the state(s) of the oject(s) that it is interacting with Zj,j = i, 1,2 ), and local time rj To generate dynamics these oject-oject interactions have to e scheduled y an update functional U (eg parallel, random, discrete event, etc) If we for simplicity assume that the update is times stepped (as in the aove LMA systems), which means that all ojects always have the same local (= gloal) time t, the dynamics of the interacting ojects in (3) is given y i = 1,,n Note that in general no explicit, closed form function F :X --+ X exists that takes the current gloal state X ( t ) = (Zl(t), * -,en@)) E x (5) *Note that we are considering fixed ojects with no selfprogrammingor mutations in the explicit rules and variales and maps it into some other state in the state space X F ( X ( t ) ) = X ( l 1) (6) + Such a function is only implicitly given through (4) Systems which can e expressed explicitly in the closed form (6) we call models Oviously, the classical dynamical systems which can e explicitly written in the form (6) are special cases of the form given in (3) and (4) This is true, ecause a system that explicitly can e written in the form ( 6 ) can e viewed as a single oject SIfrom (3)which is iterated y f11 Since there is no scheduling with only a single oject, the update functional U,ecomes the identity One of the mathematical consequences of having the dynamics of the form (3) and (4),is that (4)may not e updatale due to conflicts etween the fjj s and U Thus, a notion of simulataility can e defined [9] The connection etween the notion of simulataility and the notion of computaility can now e studied, and indeed the concepts are different, since computaility is defined through and simulataility is defined through Thus, we can have simulatale systems with noncomputale properties generated y the dynamics In addition a mathematical machine, a universal simulator (US), can e defined, which in a finite numer of steps determines whether a system is simulatale or not y descriing the conditions under which a system of the form (3) and (4)can distriute update functions over system ojects [9] Further, it can give an appropriate order of updating for the ojects, if it is simulatale, and detect where the prolems are if the system is non-simulatale Thus, the scheduling prolem naturally lives in the US Because each operation done y the U S can e interpreted as having a computational cost, the load alancing prolem also naturally lives in this mathematical machine For a more detailed discussion of some of the mathematical consequences of the aove we refer to [9] Thus, y defining dynamical systems of the form (3) and (4) an alternative and more general avenue is open We are not limited to closed form models which only keep track Q the nwner, or concentralion, of particular ojects and where novel relations and ojects cannot in a natural way e generated as a function of the dynamics (as eg in differential equations) Since the ojects in (3) and (4) are

5 explicitly represented, so are their functional properties, thus the molecular aggregates and their formation rates discussed aove all come out as oservale, emergent properties generated y the dynamical system Therefore with this approach there is no need for explicitly taking all conceivale interactions and new possile products into account together with postulates aout their formation rates By defining the ojects, the update functional, and the oservational functions appropriately, everything of interest will e generated y the dynamics Thus, we may view a simulation as an emergence engine Acknowledgements We would like to thank Kai Nagel for critical discussions during the work that led to some of the results presented here References [l] Baas NA, Emergence, Hierarchies, and Hyperstructures, Artificial Life 111,Ed CG Langton, SFI Studies in the Sciences of Complexity, Vol XVII, , Addison-Wesley, 1994 [2] Baas NA, MW Olesen, and S Rasmussen, Higher Order Emergent Structures, LANL and SFI preprint [3] Fontana W, and LW BUSS, The arrival of the fittest: Toward a theory of iological organization, Bull Math Biology, 56, 1-64 (1994) [4] Frisch, U,Hasslacher, B & Pomeau, Y, Lattice Gas Automata for the Navier-Stokes Equation, Physical Review Letters 56, (1986) [5] B Mayer, G Kohler, and S Rasmussen, Simulation and Dynamics of Entropy-driven, Molecular Self-assemly Processes, LANL and SFI preprint [6] Ostrovsky O, M A Smith and Y Bar-Yam, Applications of Parallel Computing to Biological Prolems, Annu Rev Biophys Biomol Struct, 24, (1995) [7] Rasmussen S,C Knudsen, R Felderg, and M Hindsholm, The coreworld: Emergence and evolution of cooperative structures in a computational chemistry, Physica D 42, (1990) [8] Rasmussen S and JR Smith, Lattice Polymer Automata, Ber Bunsenges Phys Chem 98, (1994) No 9 [9] S Rasmussen and CL Barrett, Elements of a Theory of Simulation, in Advances in Artificial Life, Eds: F Moran, et al, Lecture Notes in Computer Science, 929, , Springer Verlag, 1995 I

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