Abstract This paper explores the ability of molecular evolution to take control of collective physical phases, making the first decisive step from independent replicators towards cell-like collective structures. We develop a physical model of replicating combinatorial molecules in a ternary fluid of hydrocarbons, amphiphiles and water. Such systems are being studied experimentally in various laboratories to approach the synthesis of artificial cells, and are also relevant to the origin of cellular life. The model represents amphiphiles by spins on a lattice (with Ising coupling in the simplest case), coupled to replicating molecules that may diffuse on the lattice and react with each other. The presence of the replicating molecules locally modulates the phases of the complex fluid, and the physical replication process and/or mobility of the replicating molecules is influenced by the local amphiphilic configuration through an energetic coupling. Consequently, the replicators can potentially modify their environment to enhance their own replication. Through this coupling, the system can associate hereditary properties, and the potential for autonomous evolution, to self-assembling mesoscale structures in the complex fluid. This opens a route to analyse the evolution of artificial cells. The models are studied using Monte Carlo simulation, and demonstrate the evolution of phase control. We achieve a unified combinatorial framework for the description of isotropic families of spin-lattice models of complex phases, opening up the physical study of their evolution. INTRODUCTION Physical and chemical models of evolving molecular systems, both theoretical and experimental, have provided important insight into the structure of evolution (Eigen et al. 1989; Johns & Joyce 2005) . To date, however, most of the modelling of evolving chemical systems has been based on chemical reaction kinetics (Eigen et al. 1989) , sometimes in conjunction with molecular structure as in RNA landscapes (Eigen et al. 1989; Fontana et al. 1989; Schuster 2006) , with spatial effects when considered, studied in terms of reaction-diffusion equations (Boerlijst & Hogeweg 1991; Bö ddeker 1995; Cronhjort & Blomberg 1997 ). The interplay of collective self-assembling mesoscale structures (Coveney et al. 1996; Mayer & Rasmussen 2000; Nielsen et al. 2004) with the evolution of molecules has not yet been investigated in a coherent physical model. The current work aims to bridge this gap and thereby to open up the joint investigation of dynamical self-organization with a single model exhibiting coupled self-assembly and evolution. Models of amphiphilic self-assembly have been much investigated (Gompper & Schick 1991; Elliott et al. 2006) , not least owing to their relevance to cellular membranes, but although the relations between molecular properties (such as head group size and chain length) and the equilibrium phase diagram are now quite well understood, the evolution of molecules utilizing collective phases based on this relation is not. The evolution of genetic molecules with surfactant properties was proposed to be important for the origin of life (Colgate et al. 2003) . The physical connection between evolution and self-assembly in complex fluids is the topic here. INTRODUCTION Rather than starting with a simulation of a physical abstraction of a particular protocell (Rasmussen et al. , 2004 , we seek to explore the general class of evolvable multiphase systems in a physically grounded model. This work extends insights from evolving spatial reaction-diffusion systems (Bö ddeker 1995; McCaskill 1997; McCaskill et al. 2001) to such multiphase fluids, where mesoscale structures create mobility barriers and local thermodynamic reaction conditions for replicators. In terms of evolution theory, the question is how different replicators can cooperate to induce appropriate collective phases for their proliferation. The effects of mesoscale structures on molecular evolution are bound to be important because it is now well accepted that spatial localization of chemicals, even in simple reaction-diffusion systems, enables complexity in evolving systems (McCaskill 1997; Altmeyer et al. 2004) . In particular, the Gray-Scott mechanism of resource-limited cooperative replication was shown (Pearson 1993) to support growing and dividing spot patterns, reminiscent of cells, and these structures actually in turn promote the evolutionary stability of cooperative replication. In any case, the importance of cell-like mesoscale structures for biological evolution need hardly be stressed, so our goal is to reduce the model complexity to the point where the key processes and phenomena can be illuminated by simple, but still grounded, physicochemical models and simulation. INTRODUCTION Our model is intended to remain as simple as possible while retaining the essential features of an amphiphilic system under the control of a combinatorial family of replicating molecules. To this end, for the amphiphilic system, we base our model on the simple spin-lattice models introduced by Ising (1925) , Widom (1986) and Hansen et al. (1991) , which we shall however generalize to embed them in the context of a general evolvable family of isotropic models with local interactions. The Ising model will serve as a simplest reference system for highlighting the basic physics entailed in coupling collective physical phases with evolving molecules, towards life-like systems and demonstrating that important issues of collective evolutionary self-organization can be addressed in the simplest models of phase transitions. The Widom case establishes the applicability of these considerations to amphiphile systems relevant to emerging life. We will study our model in two and three dimensions: in two dimension for visual simplicity and computational speed and in three dimension for more physical realism and topological variety. INTRODUCTION In addition to the chemically simple amphiphilehydrocarbon-water system, we assume that a combinatorially complex class of molecules, capable in principle of self-replication, may also be present. These combinatorial molecules are assumed to influence the free energies of local amphiphile configurations, and consistent with this influence, their diffusion on the lattice is kinetically biased by these induced free energy changes. If we include some specific influences of the local phase on the replication or degradation processes for these replicators, we arrive at a rich thermodynamically consistent model of amphiphile phasedependent and phase-influencing replication that may serve as an elementary physical model for studying the evolutionary transition from chemical to cellular systems. Thermodynamic consistency is important if we are to ensure a physically sound model of information flow in the interplay between self-assembly and catalytic self-organization. THE EVOLVING SELF-ASSEMBLING PHASES MODEL In this section, we first introduce the basic homogeneous, spin-lattice thermodynamic models and their extensions to a family of more general isotropic Hamiltonians. We then discuss our choices in formulating kinetics (non-equilibrium treatment) for these models, before describing our main innovation of heterogeneously modulated spin-lattice systems. The local modulation of thermodynamic parameters is attributed to the presence of a family of combinatorial molecules, which are introduced as initially static, then as mobile diffusing and finally as reacting and self-replicating entities. THE EVOLVING SELF-ASSEMBLING PHASES MODEL (a) The basic Widom spin-lattice ternary model The underlying amphiphile model we have chosen is motivated by the success of Widom's spin-lattice model and its relatives in modelling the phase diagram of ternary fluids. The Widom model (Widom 1986) reduces the complexity of ternary fluids comprising hydrocarbon, water and amphiphiles to a two-state model, bringing it remarkably close to the simple Ising model, which can also be applied to fluid mixtures. THE EVOLVING SELF-ASSEMBLING PHASES MODEL The Widom model is based on a regular square or cubic lattice of sites with two states (C, K) that represent a collection of either hydrophobic or hydrophilic ends of molecules meeting at the site. In this model, all species are modelled as dimers, straddling two sites of the lattice, with hydrocarbon (oil) molecules (K, K), water (C, C) and amphiphiles (C, K) or (K, C).Owing to their prohibitively large energy, configurations, where mixtures of hydrophobic and hydrophilic ends meet, are excluded. This is reasonable for systems containing amphiphiles, where oil-water interfaces will always be mediated by amphiphiles. Although the model cannot well describe the isolated amphiphiles in water and hence the critical micelle concentration, it has proved effective in reproducing many features of the phase diagram of oil-water-amphiphile emulsions. The model is grand canonical such that both material and energy can be freely exchanged with the environment. In particular, activity coefficients are used to specify the concentration biases of water to oil and amphiphiles to water and oil. For example, changing a local 'spin' state may induce a change in the number of amphiphiles in the system. THE EVOLVING SELF-ASSEMBLING PHASES MODEL The equilibrium thermodynamics can be derived from the partition function, which is defined in box 1. The Widom model involves a mean energy difference K when the ends of the two amphiphiles meet at a lattice site, rather than two ends of oil or water dimers, reflecting differences in the interactions of amphiphiles compared with water or oil. Furthermore, the hydrophobic ends of amphiphiles can have a different energy K(1Cl) from the hydrophilic ends K(1Kl). The relative activity coefficients, z 1 and z 2 , describe the entropic cost in the grand canonical ensemble of number fluctuations away from equal occupation by water versus oil, Dn C , and amphiphile versus homophile, Dn G , respectively. THE EVOLVING SELF-ASSEMBLING PHASES MODEL (b) A general evolvable local isotropic Hamiltonian The Widom model is a particular instance of a local energy model for an amphiphilic system, assigning energies to amphiphilic spin configurations consistent with translational, rotational and reflection symmetries of space. In fact, examining the most general model with nearest neighbour interactions, with a view to later permit the evolution of arbitrary local amphiphilic interactions, we find a reduced number of distinct local spin configurations, subject to spatial symmetries depending on the lattice. Although the situation is simpler on a square 4-neighbour than a hexagonal 6-neighbour lattice in two dimensions, we have used the hexagonal lattice for some of our simulations owing to known artefacts on a square lattice. In three dimensions, the greater number of neighbours appears to allow enough configurational flexibility even for a cubic lattice. THE EVOLVING SELF-ASSEMBLING PHASES MODEL Extensions to allow longer chain amphiphiles, as in the Larson model (Larson 1988 (Larson , 1989 (Larson , 1992 , which allow further mesoscale structures such as vesicles, as well as for an explicit presentation of amphiphiles (e.g. Schick & Shih 1986) , are postponed to subsequent investigations. Extensions to include higher-order spin terms (e.g. four spin) with next to nearest neighbours, allowing the alignment costs for amphiphiles to be considered, while of interest in capturing certain properties of the phase diagram, are clearly in a later round of investigation for evolutionary models. Thus, while the Widom model is known to have certain limitations in describing some aspects of the ternary phase diagram, essentially these simple spinlattice models are known to explain the main features of the progression of phases in micellar systems with oil, water and amphiphiles (Widom 1986) . THE EVOLVING SELF-ASSEMBLING PHASES MODEL In a general isotropic (nearest neighbour) lattice model of this kind, the energies are determined by the interactions between adjacent spins and spin triples, independent of their detailed relative orientation. Allowing spatially symmetric additive contributions to the energy, up to spin triples, with no distinction between different geometric arrangements of triples, the local energy depends on six parameters, E½x Za 1 ðh 1 C h 2 s½xÞ C a 2 ðh 3 C h 4 s½xÞs sum ½x C a 3 ðh 5 C h 6 s½xÞs skew ½x; ð2:1Þ THE EVOLVING SELF-ASSEMBLING PHASES MODEL where s½x is the central spin at the lattice sitexZ fx; y; zg in three dimensions orxZ fx; yg in two dimensions and THE EVOLVING SELF-ASSEMBLING PHASES MODEL The parameters are positive integer constants depending on the lattice: for the square lattice (6,3,1) and for hexagonal (two-dimensional) and cubic (threedimensional) lattices (15, 5, 1) . THE EVOLVING SELF-ASSEMBLING PHASES MODEL For the Ising model, the only non-zero values of h are h 2 and h 4 , equal to the Ising field strength H and the spin coupling constant J. In the case of the original Widom model, the parameters h l are ðh 1 ; h 2 ; h 3 ; h 4 ; h 5 ; h 6 Þ Z K 4 !ð1; l;Kl;K1; 1; lÞ: ð2:3Þ To illustrate the extended formula, the Widom parameters can be used to calculate the local energy contribution of the central site in the configuration given in figure 1 as follows (hexagonal two-dimensional lattice): ð2:3Þ The energy contribution comes from only three triples with hydrophobic ends of amphiphiles meeting at site, as expected for the Widom parameter special case. ð2:3Þ If we completely differentiate configurations (equating the energies of states identical under rotation and reflection), then we arrive at a somewhat larger number of different energies 2 i for the general nearest neighbour Hamiltonian (not restricted to lower-order spin terms), depending on the lattice: square (12), hexagonal (28) and cubic (20) . The number of parameters is still significantly reduced compared with the number of local configurations (32, 128, 128) by the physical isotropy requirement. In addition to the interaction energy parameters, the lattice models contain parameters defining the grand canonical ensemble (either activity coefficients or chemical potentials): z 1 for Ising and {z 1 , z 2 } for Widom. While these have also been subject to genetic modulation, we defer this to later investigations. (c) Kinetics of the multiphase system Although the models are amenable to extensive theoretical analysis in equilibrium, using, for example, mean field, self-consistent field and transfer matrix techniques, and this allows a characterization of the phase diagram for the homogeneous case, our primary interest in this paper is in the kinetic perturbation of these equilibrium phases by the combinatorial molecules and so we need to incorporate kinetics. Arguably, the simplest kinetics consistent with the above partition function is a Monte Carlo (MC) model (Metropolis et al. 1953) with single spin flips resulting in a free energy change DF 1 being accepted with the probability P 1 ZMin(1, e KDF 1 /kT ). Such a single spin flip dynamics is equivalent to collectively exchanging all halves of dimeric molecules at a lattice site. In particular, this means, for example, that spin flips can cause amphiphiles and then water to appear in the middle of an oil droplet. This makes some sense for an open system in two dimensions, if we imagine the exchange of molecules with an off-surface reservoir for example, but is less easy to justify in three dimensions, unless we view the system to be in continual turnover in connection with reservoirs of dimeric molecules of the various types. F = E -TS Z = e -F/kT Restricting the dynamics to interchange of neighbouring pairs of spins (Morawietz et al. 1992) would conserve the number of hydrophilic and hydrophobic molecule ends, but not the number of amphiphiles. The kinetics with such a semi-conservative dynamics tends to be much less efficient in equilibrating the system, and so we have opted for the more efficient dynamics in this work, while aware of its limitations. F = E -TS Z = e -F/kT For the single spin flip MC dynamics, the resulting energy difference must be computed for the central site and its nearest neighbours. The change in the number of oil, water and amphiphilic dimers induced by the spin change must be computed only at the central site to determine the entropic cost of the transition associated with the activity coefficients. The resulting free energy differences ( box 1) can be computed efficiently. The traditional spin formulation can be replaced by a neighbour spin table lookup, enabling the physical transition probabilities with detailed balance to be considered in the broader context of stochastic cellular automata update rules. In particular, the traditional additive energy contributions from combinations of small numbers of spins can be extended to nonlinear combinations of spins of higher order than the nonlinear triplet terms. F = E -TS Z = e -F/kT (d) Combinatorial energetic variation coupled to the multiphase system Owing to the ephemeral nature of amphiphiles in the Widom model, it is not productive to try to attach combinatorial information about interaction properties to amphiphile configurations. The number fluctuations would result in a rapid loss of this information or require artificial reservoirs to keep track of the dynamics of hidden molecules. Instead, we chose to consider separate combinatorial molecules that modulate the hydrophilic interaction properties (e.g. K and l or z 1 and z 2 ) at a lattice site, thereby influencing the amphiphile system, but not subject to rapid number fluctuations. This has the additional advantage of separating the time-scale for combinatorial change from the relaxation time of the amphiphile system. In the discussion below, we will also refer to these combinatorial molecules as templates or replicators. Portions of these molecules encoding parameters that specify the modulation of the amphiphilic interactions (e.g. K and l) at a given lattice site are referred to as genes. F = E -TS Z = e -F/kT A static population of such combinatorial molecules corresponds to a heterogeneous spin-lattice amphiphile system model, with fixed but spatially varying energy (and/or entropy) parameters at each site. In two dimensions, this could correspond to a surface local modulation of hydrophilicity by a solid support. The result is a generalization of the spin-glass random energy model for disordered solids (Anderson 1958 ), but is not necessarily of special interest in the context of fluid amphiphilic systems. Below, we shall therefore allow the combinatorial molecules to diffuse. Regular gradients of the genetic parameters can be employed in both two and three dimensions to scan a whole range of parameter conditions for interesting local phases. Such fixed multidimensional gradient scans have already been found useful for reaction-diffusion systems such as the Scott-Gray system (Pearson 1993) and were found useful here in tuning the starting parameters of our model. We have set up our model to allow genetic parameters to potentially influence z 1 and/or z 2 and one of J for the Ising model, K and/or l, in the Widom case, a subset of the six extended Widom parameters h l in the isotropic triple spin extension, or the 12, 28 or 20 isotropic site energy parameters 2 i (for square, hexagonal or cubic lattices, respectively) in the full nearest neighbour spin model. We typically consider a linear range of variation between maximum and minimum values. This is also the range of modulation we shall investigate when allowing template mobility and evolution as described below. Figure 2 shows two typical gradient images of a simulation of the amphiphile spin-lattice system obtained for the (non-extended) Widom model itself. If we let the simulations run long enough, and make the system large enough, the gradient portrait is equivalent to an equilibrium phase diagram. Note the transitions between homogeneous, micellar, lamellar and liquid crystalline phases which are observable in the model. The micellar interpretation is problematic in the Widom model (they are actually small emulsion droplets), which also cannot exhibit the CMC transition, because it does not allow for isolated amphiphiles. Also, note that the gradients must be very weak to be able to infer accurately, from the local phase at a location in the gradient field, what the phase of the homogeneous amphiphile system with these local parameters would be. This is especially true for gradients in the activity coefficients ln z 1 and ln z 2 , since spin configurations can appear at one location and migrate (by single spin flips) to another. Nonetheless, the gradient simulations not only give a rapid indication of the equilibrium phase behaviour of the system, but also seem to capture some transient phenomena resulting from heterogeneity, which is likely to be relevant in the context of evolution. F = E -TS Z = e -F/kT (e) Mobile combinatorial molecules If we allow the combinatorial molecules to diffuse, then we have a novel mobile-fluid phase model. However, it appears at first sight unavoidable that allowing such diffusion will break the thermodynamic consistency of the amphiphile model. The problem is that diffusion of such molecules induces free energy changes in the amphiphile system through their modulation of its local energetics, and these changes need to be reflected in biasing movement. These free energy changes must be taken into account for diffusive interchange of a pair of neighbouring combinatorial molecules, which we do using a MC weighting as above. This means that for a fixed amphiphile (spin) configuration, the free energy difference, DF 2 , on interchanging the combinatorial molecules (thermodynamic parameters) between two sites, which may have either sign, is calculated (including the energetic effects induced at neighbouring sites) and the interchange accepted with probability P 2 ZMin(1, e KDF 2 /kT ). The advantage is now that we have a very natural and thermodynamically consistent feedback of amphiphile system information controlling the microscopic mobility properties of a combinatorial family of molecules. This is in fact what is required if we are to move beyond the simple pattern formation properties of reaction-diffusion systems as proposed by Turing (1952) to include the possibility of evolution. In cell biology, this model might also be used to represent the biased diffusion of protein molecules embedded in a lipid bilayer membrane. F = E -TS Z = e -F/kT (f ) Template chemistry of combinatorial molecules Chemical reactions among the combinatorial molecules allow a dynamical self-organization of the phase behaviour of the amphiphile system to take place and, in turn, the amphiphile system can modulate the chemical reactions, both through influencing the reaction rates directly via the local phase or indirectly through the influence on template mobility and effective mutual affinities via the induced amphiphile energetics (as above). We wish to include chemical reactions that allow the proliferation and destruction of combinatorial molecules, which we shall then call replicators. Chemically, no complex biochemical apparatus is required for replication, as shown by the known experimental examples of enzyme-free template chemistry such as in von Kiedrowski (1986) , although to date only limited combinatorial complexity has been achieved without enzymatic mediation. Non-explicit cooperative effects can occur even in the simple replication mechanism mediated by the amphiphile system. F = E -TS Z = e -F/kT Simple error-prone replication (X/2X ) of a combinatorial family of molecules X corresponds to the well-studied (Eigen et al. 1989 ) physicochemical model of the quasispecies introduced by Eigen (1971) . The cooperative replication mechanism (2X/3X ) is a standard model for the problem of functional exploitation (McCaskill 1997) and its spatial resolution, which is germane to the origin of life and generates proliferating cell-like reaction-diffusion patterns (selfreplicating spots; Pearson 1993) , which have been shown to allow evolutionary stabilization of the cooperative functionality with a much simpler mechanism than that of higher-order hypercycles (Eigen 1979 ) via spirals (Boerlijst & Hogeweg 1991 ). In the current investigation, it is of primary interest to investigate the indirect cooperative effect via the amphiphile phase system without such explicit higher-order catalysis. F = E -TS Z = e -F/kT Note that detailed mechanisms of templating, e.g. via polymerization from monomers or ligation of fragments, are not considered in this work. In particular, the common (Szathmary & Demeter 1987) F = E -TS Z = e -F/kT with parabolic growth and coexistence of multiple templates is not investigated here, since we are interested in the evolution of combinatorially more diverse replicator functionality. The phenomenon of coexisting templates itself is also not so interesting, despite much recent attention, being a property of the simpler self-inhibition by double-strand formation BC X / X 2 # 2X (with or without the resource B) as shown in (Biebricher et al. 1984) , and studied for ligation (Anderson 1983 ) abstracting Blum's model (Blum 1962) . F = E -TS Z = e -F/kT To connect replication with the amphiphile system of our model, three things are necessary. Firstly, we need to translate the mechanisms into nearest neighbour configurational changes on the lattice. We consider all replicators X as existing on the sites of the amphiphile lattice, occupying a lattice site with utmost single occupancy, with default amphiphile energetics being employed in their absence. Replicators X modulate the energetic properties of the amphiphile lattice at the site they occupy. When new combinatorial species are created, they are placed at a neighbouring lattice site, overwriting the original site occupant. Secondly, we need to preserve the thermodynamic consistency of the amphiphile system by testing reaction outcomes thermodynamically. Placing a replicated combinatorial molecule at a neighbouring site, affecting some subset of local amphiphile thermodynamic parameters (e.g. zs and zs), induces a free energy change DF 3 on the amphiphile system, by the same local thermodynamic contributions described above. As described above, the reaction can be accepted or rejected via the MC probability P 3 ZMin(1, e KDF 3 /kT ). Thirdly, direct influences of the local phase on the reaction can be included optionally: the reaction takes place only if the local site is hydrophobic or hydrophilic, or only at the interface between like or different phases. F = E -TS Z = e -F/kT To prevent complete saturation, the chemical destruction of combinatorial molecules is also considered. In this work, all modulators are assumed to be destroyed at a constant rate, but at a rate that may depend on the amphiphile phase. In addition to the possibilities of specific destruction rates in either the hydrophobic (K) or hydrophilic (C) phases, we may also consider the possibility of specific destruction either at an interface (a site with an oppositely signed neighbour) or in bulk (the reverse). Although we have begun exploration of a broader range of reaction behaviours in our model, in this paper we concentrate on the simplest scenario in which not only destruction but also replication rates are not directly sequence independent. This highlights the role of the complex Evolutionary self-assembly J. S. McCaskill et al. 1767 fluid coupling in determining the evolutionary outcome. Antagonistic scenarios where cooperation between replicators has to overcome differential non-cooperating proliferation have been shown elsewhere (McCaskill et al. 2001) to be kept in check by spatial correlations in a manner similar to the neutral case adopted here for simplicity. We have also chosen the simplest independent replication mechanism without specific resources and considered just two simple scenarios where sequence independent replication is allowed either in the hydrocarbon phase or at the interface. F = E -TS Z = e -F/kT (g) Combinatorial encoding of amphiphile modulation, reaction rates and mutation In order to provide a general modelling framework, we divide the genetic effects into two classes: those affecting the amphiphile thermodynamics and those affecting reaction rates. The first class encodes both the local amphiphile energetic parameters and the two chemical potentials. The second class specifies reaction probabilities, typically just one such as for replication, and this is not investigated in this work. The thermodynamic values are encoded in binary sequences, abstracting more general combinatorial structure, and interpolate linearly between maximum and minimum values. The ranges for the signed parameters 2 (or z) are given as parameters to the simulation in the form 2Z2 0 GsD2, where s is a finite precision signed binary fraction encoded by the sequence, between K1 and 1. Sixteen bits were used per parameter for z 1 and z 2 , and for K and l in the Widom case. For the extended parameters 2 i , bit lengths between 2 and 5 were used, depending on the lattice (5 for square two-dimensional lattice and 2 for hexagonal lattice). F = E -TS Z = e -F/kT Mutation is performed as single bit substitutions with a constant uniform probability. We allow multiple mutations with exponentially decreasing probability, by repeating the mutation decision for a sequence until it fails. Note that while the extended parameters 2 i can be restricted to values equivalent to the original Widom case (yielding parameters K and l), their mutation must also be restricted in order to study the evolution within the restricted Widom model. F = E -TS Z = e -F/kT (h) Simulations and software The computer code was written jointly in MATHEMATICA v. 5.2 (Wolfram Research) and the GNU GCC C/CCC compiler, using the MATHLINK connection. This allowed algebraic computations of parameters to be interfaced with optimized run-time code and provided powerful high-level graphics and GUI handling capabilities. The simulations were performed on G4 and G5 Macintosh processors under OS X 10.4, with typical equilibrating CPU run-times in the order of seconds to minutes and evolution run-times in the order of minutes to an hour. The graphical display language SDL was used for intermediate graphical display. For example, moving from left to right along the x -axis, the z 1 value decreases from 4 to K4 at the midpoint and then increases back to 4. In this fourfold symmetry, the minimum z 1 and z 2 values are at the centre of each figure. This symmetry removes edge artefacts from the simulation and allows reproducibility to be assessed. driven (non-equilibrium) dynamics genes determine replication thermodynamics and kinetics locally modify phase thermodynamics relaxation (equilibrium) dynamics multiphase system Figure 3 . The mutual influence of the combinatorial templates and the multiphase system provides a dynamic environment for evolution. The local phase of the substrate system determines both the template mobility and the template replication rate, while the template genes locally influence the ternary system, which in an energetically neutral manner can exchange material with the environment and thus change composition. The template replication process drives the thermodynamics of the composed genetic-ternary system out of equilibrium, as each replication step is associated with an energy cost. F = E -TS Z = e -F/kT (i) Discussion: coupling of system thermodynamics and kinetics The presence of a genetic molecule locally modifies the free energy of the complex fluid due to the hydrophobic and hydrophilic (or amphiphilic) interactions and Newton's Third Law dictates that this interaction has to be symmetric so that the ternary system locally influences the genetic molecule in the same manner. If, for instance, a template is amphiphilic, it will tend to locate itself at a nearby water-oil interface if possible and have a lower probability of leaving this state. Consequently, the combinatorial molecule diffusion is phase dependent since, for example, an amphiphilic combinatorial molecule is much more likely to diffuse along an oil-water interface and much less likely to diffuse into the oil or the water phases. Conversely, the presence of the amphiphilic combinatorial molecule will influence the local energetics of the substrate to produce more local interface if possible. The genephase interaction strength can be adjusted in the model using the parameters D2 i , since these specify the maximum magnitude of sequence-dependent physicochemical properties of the combinatorial molecule on the local entropic or energetics of the complex fluid. F = E -TS Z = e -F/kT Further, it is assumed that the genetic molecules' ability to replicate and/or survive depends on their environment; for example, a hydrophilic combinatorial molecule will replicate with a much higher probability in an aqueous local environment. These catalytic genetic properties are reduced to a case-by-case study in our model: the combinatorial molecules can replicate (either at the same rate or at a rate determined by their sequence) only in a predetermined local phase: either oil, water, interface, bulk or ubiquitously. Mixed scenarios are not considered at this stage. Attention here is devoted to the neutral case, where direct sequence-dependent effects on replication rate are absent: a combinatorial molecule can only influence its survival indirectly via interactions with the complex fluid phases. F = E -TS Z = e -F/kT Since the underlying multiphase system is represented in a grand canonical ensemble, there is no synthetic cost associated with mass transport of water, oil or amphiphilic molecules into and out of the multiphase system. This means that the local energetic influence that a combinatorial molecule imposes on the complex fluid can in a metabolically neutral manner drive a local mass transport of water, oil and amphiphiles. This is equivalent to assuming that the complex fluid components are freely available or readily synthesized and do not represent combinatorially complex entities. Each combinatorial molecule thereby slightly modifies the local fluid composition and thus the local phase. This resulting phase will in turn influence the combinatorial molecule mobility, the location, as well as the replication rate (figure 3). F = E -TS Z = e -F/kT The combinatorial molecules are not represented in the grand canonical ensemble because the heredity of their information is a definitive characteristic limiting evolutionary survival. Since there is free energy exchange between the complex fluid and the combinatorial molecules, the combinatorial molecules are represented in the canonical ensemble. Since our simplified model does not explicitly include any metabolic processes, and only the combinatorial molecules are conservatively synthesized by the system, genetic replication and loss defines the non-equilibrium energetic pumping of the overall system. The overall system consists of the template molecules coupled with the ternary fluid system. Each replication event is associated with a free energy cost and this (arbitrary, but fixed) free energy is supplied to the system from outside. We can safely assume that the free energy associated with this pumping (per molecule) is much greater than the free energy associated with the relaxation and self-assembly processes occurring in the multiphase system (per molecule), as the former is associated with the formation of covalent bonds while the latter is associated with the hydrophobic effect. Evolutionary self-assembly J. S. McCaskill et al. 1769 Owing to these gene-ternary fluid interactions, evolutionary adaptation happens in two fundamentally different ways in this system: (i) in the manner that is traditionally considered, where the generated variation in the template population, due to random mutations, enables selection for templates with the fastest replication rate for a given environment, which is normally called adaptation to the local environment, and (ii) the combinatorial molecules can also be selected for as they adapt the local environment to enhance their own replication rate. This is possible because the presence of the templates modifies the local thermodynamic conditions of the ternary system, which in turn modifies the thermodynamic (and kinetic) conditions for these combinatorial molecules. Thus, particular templates can be selected for, which modify the local environment, which in turn enhance their replication rate. F = E -TS Z = e -F/kT This adaptation of the local environment is a collective (cooperative) effect (of multiple genetic molecules) owing to the way local interactions collectively induce phases in the complex fluid. Such population-dependent interactions and environmental influences are known potentially to complicate Darwinian evolution, producing outcomes other than simple adaptive optimization (including limit cycle and more complex oscillatory or chaotic evolutionary dynamics). Spatial effects then clearly play an important role in the feedback between genetic influence on the complex fluid and its influence on genetic survival. THE COMBINATORIAL SELF-ASSEMBLY DYNAMICS WITHOUT REACTIONS Before considering any reactions or evolution of the combinatorial molecules, it is important to gain some understanding of the nature of this coupled dynamical system: how it affects the amphiphile phase diagram and how it impacts on the free diffusion of modulators. This will be the subject of this section. THE COMBINATORIAL SELF-ASSEMBLY DYNAMICS WITHOUT REACTIONS Although we shall demonstrate this two-way coupling using generalizations of the Widom model and its essential precursor, the random coupling Ising model, we expect the type of phenomena we demonstrate here to be a general feature of combinatorially coupled complex fluid models. THE COMBINATORIAL SELF-ASSEMBLY DYNAMICS WITHOUT REACTIONS Note that the influence of random static, nonreacting combinatorial molecules is related to the random energy models as introduced in the study of spin glasses and other disordered solids (Anderson 1958) . The simplest case is just an Ising model with a random site energy (or single spin H parameter) or random coupling terms J. In this static case, it is known that the disorder itself can induce localization thresholds in the mobility of particles subject to this Hamiltonian, but this is unlikely to be the case if the random energies are themselves mobile. In the case of random H, the perturbations on the Ising model behaviour appear local. In the case of random J, the critical temperature itself can be shifted radically by random perturbations as shown in the gradient plot in figure 4 . In order to demonstrate the main effect, we chose reference systems with zero interaction energies. In the lower plots, static inhomogeneous perturbations of the Ising model, as applicable to a binary fluid, are shown. Note that in the absence of the fixed combinatorial molecules, there is no ordered phase at any temperature since J is set to 0. The correlated patches in the ordered regime are at fixed locations in the random J model. The figures were created with MC simulations starting with a random spin configuration. THE COMBINATORIAL SELF-ASSEMBLY DYNAMICS WITHOUT REACTIONS A similar result is achieved for the Widom model with fixed combinatorial molecules perturbing K C Z(1Cl) and K K Z(1Kl), as shown in figure 4c . Note that, it is more natural to treat these two quantities as independent random variables subject to genetic control, than K and l directly, because they represent the separate energies of the juxtaposition of hydrophilic or hydrophobic ends of two amphiphiles ends, respectively, compared with water or hydrocarbon. Although these plots are noisy, being subject to the uncorrelated perturbations of the combinatorial molecules, more correlated perturbations induced by the combinatorial molecules can have a stronger influence. THE COMBINATORIAL SELF-ASSEMBLY DYNAMICS WITHOUT REACTIONS If we now allow the combinatorial molecules to diffuse, as befits disordered fluids, in contrast with solids, then a number of new phenomena emerge. Recall that the random walks must be biased by the energetic changes induced in the binary (Ising) or ternary (Widom) fluid for thermodynamic consistency, as discussed in §2. Consequently, the mobile combinatorial molecules can both influence the phase of the complex fluid in which they float, and their spatial arrangement and mobilities can be influenced by the induced phases. In figure 5 , we show this coupled phase behaviour for the Ising model coupled to mobile elements, which randomly perturb the coupling constant J from 0. This describes the way in which a random population of combinatorial molecules with varying degrees of amphiphilicity interacts with a homogeneously mixed twocomponent fluid to induce mesoscale structuring of the fluid and combinatorial sorting and clustering of the combinatorial amphiphiles. Note that the combinatorial molecules themselves may experience effective, attractive or repulsive interactions allowing aggregation, for example, induced by their impact on amphiphile energetics. We note that such examples of noise-induced order may assist the onset of cooperative action by populations of genetic elements, through correlating collocation with function autonomously. THE COMBINATORIAL SELF-ASSEMBLY DYNAMICS WITHOUT REACTIONS A similar non-random clustering of the diffusing combinatorial molecules, interacting with a Widom ternary emulsion system of hydrocarbon, water and surfactant, occurs as shown in figure 6 . So, in addition to the vertical gradient, we also show a horizontal gradient in the activity coefficient of amphiphile (z 2 ). Near z 2 Z0, at the left and right extremities of the plot, the higher temperature phase is the mixed system, whereas at lower amphiphile activities, in the centre, the higher temperature phase is the separated fluid. The spatially resolved inverse mobility plot in figure 6b confirms the structured feedback on the combinatorial molecules through their perturbation of the complex fluid. The values shown in figure 6b are calculated as the free energy cost in units of kT for an interchange of two neighbouring amphiphiles in the vertical direction. The plot looks similar for interchanges in the horizontal direction. The mobilities of the combinatorial molecules become strongly correlated in spatial clusters of molecules having similar interaction parameters below the transition to the higher temperature phases. THE COMBINATORIAL SELF-ASSEMBLY DYNAMICS WITHOUT REACTIONS In figure 7 , we show the corresponding free energies that drive the equilibration process for the Widom model case shown in figure 6. Note that equilibration is essentially complete after 1000 MC updates per site. It is worthwhile to observe that the combinatorial molecules relax to lower the free energy of the system, at the expense of the configurational free energy associated with the ternary fluid, consistent with the equilibrium state containing more structure than in the absence of the combinatorial molecules. THE COMBINATORIAL SELF-ASSEMBLY DYNAMICS WITHOUT REACTIONS In summary, in this section, we have shown that mobile combinatorial molecules can both modulate the phase of complex fluids and themselves be ordered by this interaction. It is significant that random perturbations by combinatorial molecules can induce phase transitions to more ordered phases of complex fluids (as in noise-induced order). Note that although static random energy and random coupling models have been studied in solid-state areas such as spin glasses, the equivalent for a multiphase liquid system with mobile elements does not appear to have received attention until now. The ability of molecular interactions with a complex fluid to provide structured pattern formation in a system of combinatorial molecules introduces a rich range of novel pattern-forming mechanisms to act as a substrate for evolution. This will be explored in §4. THE EVOLVING REPLICATOR-AMPHIPHILE SYSTEM In this section, we shall explore the coupled evolution and self-assembly which our model captures. This is accomplished by allowing reactions to take place among the combinatorial molecules, in particular, replication, which enables inheritance and subsequently evolution. The simplest case is that of the Ising model as applicable to a simple binary fluid, as discussed in §3, which we shall examine first. This exemplifies the character of evolution coupled to phase control in a thermodynamically consistent way. The second case is the Widom model, and since it incorporates a combinatorial treatment of amphiphiles, this of course is more relevant to the self-organization of protocells. Both models are special cases of the general isotropic local Hamiltonian introduced in §2, which will be investigated in the future for more general evolution. THE EVOLVING REPLICATOR-AMPHIPHILE SYSTEM Consider firstly the evolution of a population of combinatorial molecules which can only replicate in an organic phase, but can influence the coupling constant J, determining the interaction energy between the organic molecules and water, in a sequence-dependent fashion. Since non-zero coupling constants can support collective phases, it is conceivable that templates with sequences for appropriate values of J can cooperate to produce a local oily phase and thereby increase the replication probability and survival of these templates. As in §3, we set up a temperature-gradient environment (from 0.25 to 2.25), with the environmental coupling coefficient J 0 Z0, so that the initial fluid is in a well-mixed state containing equal amount of water and organic material. The range of combinatorial variation is set to G1. There is no combinatorial variation in replication probability or loss rate, which are set to 0.5 and 0.1 per MC update (unit time). We use simple replication and the binary landscape described in §2 and a single mutation rate of 0.01 (per replication event, with multiple mutation events per replication allowed). The results are shown in figures 8 and 9. Note that at low temperatures, the steady-state population supports some 'parasitic' combinatorial molecules, which do not contribute in full to maintaining the oily phase (figure 8e). THE EVOLVING REPLICATOR-AMPHIPHILE SYSTEM The evolution of combinatorial molecules capable of acting as or producing amphiphilic molecules will be the subject of the remainder of this section. This is of some significance to the evolution of cellular life, as suggested by the experiments on self-reproducing surfactant systems (Luisi & Varela 1989) . Already in the Ising model simplification, some useful insights can be gained, so we present an evolutionary simulation before proceeding to our main result on the Widom model. If replication of template molecules can only occur at the interface between hydrophobic and hydrophilic phases, an assumption relevant to the Los Alamos protocell model , then the situation changes dramatically from the replication in oil scenario shown above (figures 8 and 9). THE EVOLVING REPLICATOR-AMPHIPHILE SYSTEM We commence with a finite environmental value to the coupling constant JZ0.5, so the system is phase separated at nearly all temperatures in the gradient simulation (T from 0.5 to 1; figure 11a ). We allow a population of combinatorial molecules to influence the coupling constant as in the previous example. Although we could have attained more rapid evolution with the binary sequence landscape used in the previous case, we have used a unary encoding of the coupling constant here (30 bits), which serves to illustrate a landscape where significant functionality is rare and many mutations from random sequences are required to reach this. THE EVOLVING REPLICATOR-AMPHIPHILE SYSTEM The evolution shows increasing adaptations of the combinatorial molecules to produce interface via negative J values and thereby an emulsion of the oilwater system. Initially, the population is localized near the macroscopic interface, with minor departures determined by the relative magnitudes of diffusion and loss probabilities. Interface forms more easily at high temperatures, so the time course (figure 10) shows a successive occupation towards the low-temperature centre. The steady-state evolved population occasionally completely fills the space, but predominantly a small phase-separated region at low temperature remains (figure 11c). On average, five adaptive mutations are required to achieve the coupling values of the evolved population, with more negative values being adapted in the central low-temperature region (figure 11e). Compare this with the neutral case, where local spatial correlations induced by the replication-diffusion system are visible (figure 11f ). The free energy traces in figure 10b show that the evolution drives the system steadily, but nonmonotonically, to a higher free energy state with about half the free energy in the genetic modulation. THE EVOLVING REPLICATOR-AMPHIPHILE SYSTEM The final example discussed in this section demonstrates the evolutionary adaptation of replicators' modulator genes to enhance their proliferation via genetic control of self-assembled amphiphile structures. The example is not just selection from among the variants present in an initial population, figure 4d ) . A vertical temperature gradient is applied, here ranging from 0.2 in the centre to 1.2 at the top and bottom of the plots, visualizing the phase diagram. A phase transition is observed at a temperature of ca 1.0 between a homogeneous mixed fluid phase at high temperature and an ordered mesoscale phase, consisting of three mesoscale structures: regions of hydrocarbon, water and liquid crystal. The MC simulations on a 256!256 square lattice were run until equilibrium, constant free energy, was attained. but a sequence of at least two separate adaptive events. The first occurs some time after the simulation starts, when a colony of replicators first successfully adapts to the watery phase. This adaptation changes the water phase by creating lots of micelles, thus maximizing interface (thus, maximizing reproduction). The second, later adaptation arises independently in a number of new colonies that learn to survive in the oily phase by creating reverse micelles, thus maximizing interface. The timing of these adaptations indicates that they were not present in the initial population (or were present but did not survive). THE EVOLVING REPLICATOR-AMPHIPHILE SYSTEM The example uses an environment created by a gradient of z 1 ranging from K2.7 to 2.7, which creates two bulk phases (water and oil) with a complex interface between them. The replicators evolve different adaptations to survive in these niches, as well as at the interface. Replicators are chosen to replicate only at the interface. Other thermodynamic parameters are kTZ0.2, z 2 Z1.3, KZ0.25 and lZ0.01. Figure 12 illustrates the approach to equilibrium behaviour of the amphiphile system with no replicators present in figure 12a,b. In figure 12b , we see the watery phase on the right (white), the oily phase on the left (black), complex phase in the middle with a few micelles (black dots in white), a few reverse micelles (white dots in black), and various bilayer and inverted bilayer structures that create a large amount of interface (amphiphiles, at the border between white and black). Figure 12c illustrates the asymptotic effect of the successfully evolved replicators on that equilibrium. Note that the replicators have changed the selfassembled structures by filling the watery phase with micelles and the oily phase with reverse micelles. Comparison of figure 12b with figure 12c dramatizes the effect of coupling replicators and self-assembled structures. The replicators have changed the Corresponding plot for a neutral gene (one which has no effect on amphiphilic spin dynamics, but undergoes the same replication process as those that do). amphiphilic structures to create much more interface, which they need to reproduce, thus evolving a coupling with the amphiphiles that produces a favourable local environment for themselves. Figure 13 illustrates key statistics measured as the replicators evolve. Note that the population shows signs of making two significant adaptations, the first around time-step 300 and the second around time-step 800. The timing of the first adaptation (around time-step 300) corresponds to the onset of a steady fall in the mean value of gene 1; the corresponding characteristic rise and fall in the variance in gene 1 and the rise of the replicator concentration to its first plateau. The timing of the second adaptation (around time-step 800) is confirmed by the slight final bump in mean value of gene 1 in figure 13d when the new oily living subpopulations grow and raise the mean gene 2 value; this
{'paper_id': '20259320', 'title': 'Evolutionary self-organization in complex fluids', 's2_url': 'https://api.semanticscholar.org/CorpusID:20259320', 'source': 'S2ORC'}