GENERAL DISCUSSION Dr. P. Ortoleva and Prof. J. Ross (MZT)said We have investigated the problem of the establishment of gradients of concentrations in a chemical system far from equilibrium which is originally homogeneous.' For this purpose we chose a volume surrounded by a membrane and that composite system is immersed in a bath of fixed concentrations. Note that the boundary condition across the membrane for each M -Memb raw-> M distance FIG.1. concentration is a flux proportional to the concentration difference across the mem- brane. For cells this is likely to be a physically better condition than one of constant concentration or constant flux. We formulated a general theory for symmetric and asymmetric perturbations and then derived for a model system the equations for the symmetric and asymmetric stable states.Thus we showed that from an initially stable symmetric state in a system immersed in a bath of constant concentrations (dotted line in the figure) an asymmetric stable state (solid line) evolves in certain ranges of kinetic and transport coefficients and bath conditions. A different problem of spatial order is that of Liesegang rings referred to earlier in this Discussion. A theory and some experiments on this topic have been presented by Flicker and Ross2 Dr. A. Babloyantz and Prof. G Nicolis (Brussels) (partly communicated) :To our knowledge the paper by Ortoleva and Ross refers to a discontinuous system of two compartments subject to a flow of matter from the outside. One cannot really refer in terms of " boundary condition " to the flux across the external " membrane " as this flux is incorporated into the kinetic equations themselves.Asymmetric states of concentration under similar conditions were first obtained by Prigogine and Lefe~er.~ In spite of its interest and its impact as the first model for dissipative structures a discontinuous two-box model suffers from the inconvenience of imposing artificially a wavelength on the system. This point is discussed in detail in Prigogine's paper at the second Versailles conference From Theoretical Physics to Biology (Versailles 1969). When diffusion is taken into consideration either in a P. Ortoleva and J. Ross Biophys. Chem. 1973 1 87. M. Flicker and J. Ross J. Chem. Phys. 1974,60,3458.I. Prigogine and R. Lefever J. Chem. Phys. 1968,48 1695. 263 GENERAL DISCUSSION continuous formalism or by dividing the space into at least 3 cells then the system can evolve to a structure whose wavelength is no longer imposed but is instead determined by the system's parameters the boundary conditions and the size. Under these conditions it has recently been shown that polarity can appear spontaneously in an initially homogeneous morphogenetic field. Dr. P. Ortoleva and Prof. J. Ross (MIT) (communicated):Prigogine and Lefever' showed with a model reaction in a two-box enclosure that gradients of chemical species may be established under the conditions that these species (X Y)do not inter- change with the surroundings. We showed that in a similar system (eqn A.7 8) a gradient of species can be established under more restrictive conditions such that the species for which the gradient is formed may exchange with homogeneous surroun- dings.2 Thus despite the driving force by the constant concentrations in the sur- roundings (bath) to erase any gradient such a gradient arises. This is of importance in those cases of morphogenesis taking place in homogeneous surroundings for which the morphogens may move across the boundary of the system. The two-box model is indeed a simplification but nonetheless is related to the continuum partial differential reaction diffusion equations with boundary conditions as may be shown by a mode-mode coupling theory. The lowest order approximation in that approach yields a cosine distribution in position with an amplitude obeying a nonlinear equation which is essentially that for the two-box model.Dr. J. S. Turner (Azutilz Texas) said In Section 2 of the paper by Goodwin a model for membrane excitability is proposed with the qualification that full experi- mental justification for a phase transition interpretation of the phenomenon is not yet available. Insofar as the underlying chemical kinetics of an appropriate model for membrane excitability is found to exhibit multistationary states then interpretation as a phase transition may well be accurate. The question of phase transitions in systems for which macroscopic chemical kinetics predicts several stable states has been studied recently by Lefever Prigogine and myself.Analyzing spontaneous localized fluctuations in a simple chemical model we have verified the existence of metastable states in the coexistence region and have obtained the minimum coherence length L for spontaneous fluctuations necessary to induce a transition from such a state. (For details see my discussion remark following the paper of Nicolis and Prigogine in this Symposium.) In this view for example the phenomenon of latent excitability referred to at the end of Section 2 would correspond to naturally occurring fluctuations having a characteristic spatial extent L < L, a consequence possibly of physical constraints on membrane-related processes. Dr. A. Babloyantz (Brussels) said In his paper Goodwin considers the sudden appearance of spontaneous action potential during the regeneration of the unicellular marine alga Acetabularia.But the differential equations constructed to describe his model are linear ones and cannot give a spontaneous generation of inhomogeneity in a given physico-chemical parameter starting from a homogeneous situation. The inhomogeneous result found by the author is due to his special choice of the so-called pick-up function. In particular for the special case of Acetabularia any theory has to explain how in anucleated fragments one can have a generation of polarity giving rise either to one cap formation at one end of the fragment or two caps at either end I. Prigogine and R. Lefever J. Chem. Phys. 1968 48 1695. P. Ortoleva and J. Ross Biophys. Chem.1973. 1 87. GENERAL DISCUSSION of the fragment. Can the author account for these experimental facts? And if so how is the number of caps determined within his linear theory? Dr. B. C. Goodwin (Sussex) said Babloyantz and Auchmuty questioned whether the model of wave propagation and gradient formation on a membrane was spatially uniform initially and whether it is dynamically stable. It is evident from fig. 1 that all the molecular components of the model are distributed uniformly in the membrane so that there is no initial asymmetry. Once a wave begins to propagate from some point however there is first dynamical asymmetry (i.e. at any point in space one process is followed by another in a defined manner as determined by the pattern of interactions described by the kinetics of the reaction scheme) and subsequently spatial asymmetry in the bound metabolite arising from the dynamical asymmetry together with wave propagation.This leaves open the question how the wave origin is determined which was done deliberately in order to cover the various cases to which the model could be applied. These include (1) an external environmental event determining where the symmetry-breaking wave is initiated as in the case of light acting upon a fertilized Fucus egg; (2) an initial gradient of some kind which establishes the wave origin as in an amphibian embryo or a decapitated Hydra; (3) a system with neither initial polar order nor an environmental asymmetry which can break symmetry spon- taneously as a result of spontaneous fluctuations which exceed the level for initiation of the wave.In this instance more than one wave origin can arise so that com- peting organising centres may result as occurs in bipolar Acetabularia segments. The model is applicable to all these instances but it does not of course explain them. Regarding stability the basic equations of the model as described for example by Goodwin are necessarily stable when P(x,t)is a propagating delta function (the case analysed) because a finite amount of substance is produced over any finite spatial region of the membrane. Thus there cannot in this instance be any question of instability and the problem considered is simply the spatial distribution of the morphogen resulting from the passage of a wave along a membrane.Dr. J. F. G. Auchmuty and Prof. G. Nicolis (Brussels) (partly communicated) In their paper Balslev and Degn appear to overlook the effect of boundary conditions on the solutions of reaction-diffusion equations. Their conclusions that (D,/D,) > 1 is a necessary condition for obtaining an aperiodic instability and that in domain I of fig. 2 the system will necessarily evolve to a uniform limit cycle only hold under special boundary conditions. An analysis of the one-dimensional trimolecular model in the case where the concentrations are held fixed and equal to their uniform steady state values at the boundaries (see the paper of Nicolis and Prigogine at this Symposium) shows that the first instability may give rise to a space-dependent and time-periodic structure.More-over such structures arise before aperiodic solutions provided where we are using the notation of Nicolis and Prigogine. Here D,,D are the dif- fusion coefficients of X (the ‘‘ self-activating ” component) and of Y. In particular D,/Dx need not be greater than 1 to lead to an aperiodic stability when A is suffi- ciently small. B. C. Goodwin Theoretical Physiology in Selected Topics in Physics Astrophysics dnd Bio-physics. ed Laredo and Juristic (D. Reidel Publ. Co. 1973) pp. 381-420. GENERAL DISCUSSION The reason for this is the fact that linear stability analysis for one dimensional systems of length Z with fixed concentrations at the boundaries allows modes of the form for n = 1,2 . . .. The state with n = 0 is now excluded because of the boundary conditions and so one gets these different results.Fixed concentration boundary conditions may arise if the reaction takes place inside semipermeable membranes and there is a reservoir of reactants outside. Prof. D. Thomas at Compihgne has been conducting experiments with these boundary conditions. Dr. A. Winfree (Indiana) said Balslev and Degn point out that in the linear approximation near a homogeneous stationary state propagating waves cannot occur in a 2-component reaction/diffusion system unless it oscillates while well stirred (i.e. with V2 term omitted eigenvalues are now real) and that initiation of such spatiaZ instability requires that the diffusion coefficients be unequal. As usual with linear systems one must enquire how much non-linearity or how great an excursion from the stationary state is required to admit the forbidden phenomena.I submit by way of answer only an isolated example but it is an in- structive one I think aA -= V2A-A-B+H(A-0.05) at aB -= V2B+A/5 at where H(x) = 1 if x > 0 or 0 otherwise. It will be noted that the diffusion coeffi- cients are equal and the equation is piece-wise linear the two pieces meeting along A = 0.05. In the well-stirred case (V2 terms omitted) the stationary state (0,O)is a unique global attractor with negative real eigenvalues -0.72 -0.27. Yet this equation’s stable solutions (depending on initial conditions and boundary conditions) include not only the homogeneous stationary state but also propagating plane waves and concentric circular waves and spiral waves polarized to rotate clockwise or counterclockwise (unpublished numerical computations).Prof. I. Balslev and Dr. H. Degn (Denmark) said In answer to Auchmuty and Nicolis we remark that in our paper we have emphasized that the boundary conditions we wish to consider are the ones a chemist will always assume if nothing else is ex-plicitly stated namely the ones of “an ordinary vessel with non-reacting walls ”. We have abstained from discussing the consequences of other boundary conditions such as fixed concentrations at the walls because we would feel obliged to discuss also the operational meaning of such boundary conditions. We have not been able to think of any operations which will assure fixed concentrations at the walls.In answer to Winfree we affirm that the hypothesis of the existence in certain reaction systems of spatial instability towards infinitesimal perturbations is a fun- damental one. The aim of our paper is to contribute to the clarification of the con- ditions required for such spatial instability. The interesting observations of propagating patterns presented by Winfree in this symposium seem to be dependent on Jinite perturbations. Likewise Winfree’s example given in his comment has a globally stable steady state. Therefore a departure from the corresponding homo-genous state can be brought about only by ajinite perturbation. GENERAL DISCUSSION Dr. R. G. Gilbert and Mr. S. McPhail (University of Sydney) said In theoretical discussions of reactions which produce spatial oscillations an obvious point which is frequently overlooked is that a stable final stationary inhomogeneous state I/~(x) is required in addition to an unstable initial state.In a one-dimensional system described by a$pt = F(+) +Da*+/aXz (1) (where t) is the usual vector of concentrations etc.) then t,V is defined by putting the right-hand side of (1) equal to zero. An indication of the stability of +I is that the real parts of the eigenvalues of the matrix aF/a$I be negative for all x. We have written a general program for numerically determining $I requiring the solution of coupled differential equations with double-ended boundary conditions (which as the equations are very badly conditioned poses considerable numerical difficulty).esting this program over a small range of x on the Brusselator mechanism we found not only the solution derived by Lefever by complete numerical solution of (I) (which was of course stable) but also several unstable solutions one of which was asymmetric and hence relevant to Babloyantz’ previous remark. We feel that the search for stable stationary inhomogeneous states will prove most useful in under-standing morphogenesis. Dr. A. Babloyantz and Dr. J. Hiernaux (Brussels)(partry communicated) :We wish to present a mechanism explaining the spontaneous onset of polarity in an initially homogeneous morphogenetic field. Ever since Turing’s * important work it has been known that a reaction diffusion system of two or several morphogens may account for many phenomena of morphogenesis.In particular the spontaneous formation of inhomogeneous patterns was viewed as the result of the instability of the homo- geneous state triggered by random disturbances. However due to the particular boundary conditions used by Turing the patterns he obtained did not present any polarity. This was also the case for all subsequent work on dissipative struct~res.~-~ Consider a reaction-diffusion system of two morphogens nz and m2 described by the following kinetic equations fand g describe chemical reactions which must obey kinetics similar to those proposed by Turing. We impose zero flux at the boundaries of the system. Then it can be shown 6* that for a proper choice of parameters there is a critical length Lo below which the system remains homogeneous.R. Lefever J. Chem. Phys. 1968,49,4977. A. M.Turing Phil. Trans. R. SOC.B 1958,237 37. P. Glansdorff and I.-Prigogine Thermodynamics of Structure Stability and Fluctuations (Wiley-Interscience New York 1971). I. Prigogine and R. Lefever J. Chem. Phys. 1968,48 1965. J. F. G. Auchmuty and G. Nicolis Buff. Math. Biof. 1975. A. Babloyantz and J. Hiernaux Science 1975 to appear. A. Babloyantz and J. Hiernaux Bull. Math. Biol. 1975. GENERAL DISCUSSION When the length of the system is greater than Lo a one-wavekngth structure appears in the system and is of the form seen in fig. 1 which presents a polarity. For higher values of L there appear structures of several wavelengths.72 m cell position FIG.1. Dr. B. L. Clarke (AZberta) said Balslev and Degn have demonstrated the desir- ability of being able to study reaction-diffusion instabilities which cannot occur in systems with too few reactants. I would like to indicate how the steady state stability of large reaction-diffusion systems can be determined. The stability analysis of large realistic chemical systems is greatly simplified by the fact that the rate constants usually range over many orders of magnitude. The matrix elements of eqn (2) and (11) (of the paper by Balslev and Degn) are then also of different orders of magnitude and the equations in the Appendix can be simplified to a few dominant terms. For large reaction systems this simplification makes an otherwise unwieldy algebraic problem tractable.Since the stability domains are determined by the signs of various polynomials the curves separating the stability domains have approximate equations which state that two possible dominant terms of opposite sign are equal. Such equations are always hyperplanes when plotted logarithmically in the parameters. From this it follows that the stability domains of a reaction-diffusion system are asymptotically convex polyhedral cones in the log- arithmic parameter space. This approach is only valid when the parameters cannot change sign. If each of the quadrants in fig. 1 and 2 is mapped into the space with coordinates (logjcxl loglp() the hyperbolas and lines through the origin will be mapped into straight lines and all the stability domains (except region I1 of fig.2) will appear as convex wedges with distortions near their apexes. The cone-like structure of the stability domains breaks down near loglctl = loglpl = 0 because the parameters in eqn (2) are then all of similar magnitude. For chemical networks it can be shown that a complete breakdown of this approximation only occurs when all reactions have the same steady state currents and all evolving reactants have the same steady state concentration. The conditions for even a partial breakdown of this approx- imate approach are not expected to be encountered often. If the stability analysis is carried out for a particular reaction network rather than for the general matrix as in eqn (2) the parameters in the matrix will all be of the form nj/c where c is a steady state concentration and j is a current and n is a small integer which comes from kinetics and stoichiometry.If diffusion is included the parameters Dk2 also occur. Since these parameters are all positive the stability B. L. Clarke J. Chem. Phys. 1974,60,1481. GENERAL DISCUSSION domains in the analogue of fig. 1 are only of interest in the positive orthant and only one mapping into the logarithmic space is necessary. Furthermore each convex cone of instability can be related to the network features which produce it. The dominant term in the stability polynomials which is responsible for destabilizing a network is a product of matrix elements and can be factored into a set of feedback loops which are important in the instability.Thus each cone of instability is caused by recognizable combinations of feedback loops and the stable regions adjacent to each cone are stabilized by other recognizable network features. Dr. M. Herschkowitz-Kaufman (Brussels)said In the paper by Balsev and Degn it is stated that the existence of multiple stable patteriis is not yet established for a reaction-diffusion system. I should like to emphasize that the multiplicity of stable steady state structures has now been firmly established both by computer simulation and analytically. Consider the trimolecular reaction scheme. In a bounded medium subjected to zero flux or fixed boundary conditions computer simulations have shown the existence beyond a certain distance from the instability point of afinitr space (arbitrary units) space (arbitrary units) (4 (b) L J 0 space (arbitrary units) (4 FIG.1.-Steady state profiles of X obtained for the same values of the parameters but different initial conditions.The curves have been established for zero flux boundary conditions. Numerical values used A = 2 L = 1 B = 4.6 D = 1.6~ D = 8.0~ multiplicityof spatial steady state dissipative structures each one having its own domain of attraction as a function of the initial conditions. This is illustrated on fig. 1. Each one of the three different stable structures observed corresponds to a different G. Nicolis Proc. AMS-SIAM Symposium on Appl. Math. vol. 8 (American Math.SOC. Providence 1974). GENERAL DISCUSSION wavelength and amplitude. They emerge from the honiogeneous steady state for the same values of the parameters of the system but different initial conditions i.e. depending on the sign location and number of initial perturbations. This phenome- non can be inderstood on the basis of a bifurcation analysis which can provide an analytical approximation for the new steady state solutions bifurcating from the n FIG.2.-Real part of the eigenvalues wIIcorresponding to the unstable modes emerging from the thermodynamic branch for different values of B (zero flux or fixed boundary conditions). The other parameters have the same values as in fig. 1. homogeneous steady state not only in the neighbourhood of the first instability point but in the neighbourhood of any marginal stability point.2 However at the present time the stability of the successive bifurcating solutions cannot be guaranteed a priori.Computer simulations indicate that those solutions are stable which correspond to the fastest growing unstable modes according to the linear stability analysis (i.e. to the space(arbitrary units) space(arbitrary units) (4 (b) FIG.3,-Degenerate steady state dissipative structures obtained for the same values of the parameters by a single initial perturbation of the homogeneous steady state of the same strength but opposite sign. The arrows show for some points the sign of the perturbation leading to the corresponding spatial distribution. The numerical values of the parameters are the same as in fig.1. The homogeneous steady state concentrations X = A Y = B/A are maintained at the boundaries. first few successive bifurcations). This is seen explicitly in fig. 2 where n = 7 8 and 9 appear indeed as the leading modes for the numerical values corresponding to fig. 1. An additional mechanism of multiplicity operating even near the first bifurcation point is degeneracy as appears from the analytical expression (3.4) presented in the G. Nicolis and J. F. C. Auchmuty Proc. Nut. Acud. Sci. 1974 71 2748. GENERAL DISCUSSION 27 I paper by Nicolis and Prigogine. Fig. 3 gives a typical example of such degenerate states obtained by computer simulations. Dr. P.Hanusse (CNRS Talence) said I wish to mention to Balslev that we have established several theorems on the occurrence of instability especially in two corn- ponent systems.* The main result is that in a system with two components when only mono and bi-molecular steps are involved whatever the form of reaction scheme 1 no unstable node or focus may occur so no limit cycle type oscillation 2 when the system is stable with respect to homogeneous perturbations that is stable when stirred it can not be spatially unstable. Consequently models with two components can lead to oscillations only if they contain tri-molecular steps as does the Brussellator. Dr. J. F. G.Auchmuty (Brussels) said I would wish just to mention to Ross that for single component reaction-diffusion systems one often gets the single front propagation described in his paper in which one phase annihilates the other.In multicomponent systems however other phenomena besides coexistence and pro- pagating fronts can occur. These phenomena include the existence of fronts whose propagation velocity is not constant and of unstable propagating fronts as well as the nonexistence of such fronts. In such examples the boundaries between the states can become very com- plicated (as for example often occurs in the Stefan problem) and for certain model systems one can even get explosions when two phases are mixed in the manner described in this paper. Even the very concept of boundary between two phases is very difficult to define when the steady state solutions are space-dependent. Prof.J. Ross (MZT)said In the concluding paragraphs of our paper we comment on the agreement of two methods for analyzing aspects of nucleation one based on solutions of the time-dependent macroscopic (deterministic) equations the other on the steady state probability distribution obtained from a master equation. Since submission of this article we (K. Kithara H. Metiu and J. Ross) have found reasons for such agreement. In brief we have derived by path-integral methods solutions of the master equation for " curl-free " reaction systems and a certain form of the transition probability which show that the deterministic equations yield the trajectory for the most probable dynamic evolution. Hence the evolution of such non-linear system in the direction of increasing probability is on the average given by the deterministic equations.Dr. P. Fife (Tucson Arizona) said These remarks extend the results given by Nitzan Ortoleva and Ross in their Section 2. 1. Consider the case of a single dependent variable tl/. Let $*(A) and t.,bl(A) be the two stable stationary states and P,Hanusse Compt. Rend. l973,277C,(63 1247.l972,247C ,; GENERAL DISCUSSION If there is onIy one stationary state (unstable) between $o and 11/1 then the critical values R,(2.4) can be found from the simple relation l* J(RM) = 0. If there is more than one stationary state between $o and $ 1 then the extra condition F(IC/ nM)d$ > 0 for $O(AM) < k < #l(RM) 'hO(AM) has to be imposed. In the case of more than one dependent variable apparently there is no such simple criterion available for determining lbM.2. Again consider one dependent variable but an inhomogeneous medium with the constraints A held fixed. Thus dependence on x replaces dependence on 2. Define t,bo(x),$l(x) and J(x) as above. The coexistence problem is to find a solution of D9+F($ x) = 0, dx2 ~ with exhibiting a phase transition from valuesnear t,bo(x)to values near t,bl(x) as x crosses some transition region. If D is small a singular perturbation analysis '9 may be applied to determine the conditions for such a solution to exist. Suppose there is a value x* such that J(x*) = 0 and J(x) changes sign as x passes through x* (also assume an extra condition like that above if there is more than one stationary state between $o and Then a phase transition may occur in an interval centred at x* with width of order D*.Furthermore a heuristic argument shows this coexistence solution to be stable provided that J(x) > 0 for values of x "in phase lcfo "; J(x) < 0 for values of x "in phase ". This contrasts with the homogeneous case treated in Ross's paper in which a small perturbation of 1 will start the front moving with consequent annihiIation of one of the phases. If J is negative for some finite interval a < x < b only then can exist in that same interval with transitions to t,bo at both ends. 3. Consider now the case of two dependent variables u and v and a homogenous medium.2 Assume the diffusion coefficient D of u to be much smaller than that of u which we set equal to 1.The steady-state equations are d2u D -= f(zr 0) dx2 d2v -= g(u u). dx2 Suppose the curve f = 0 is S-shaped with t,bo and $1 the two stable branches P. Fife J. DiY. Qn. 1974 15 77. See esp. pp. 102 and 103. P. Fife J. Math. Anal. Appl. 1975 to appear. 'P. Fife and W. M. Greenlee Uspeki Mafem. Nuuk SSSR 1974 49 103 ; also to appear in Russ. Math. Surveys. A. B. Vasil'eva and V. A. Butusov Asymptotic Expansions of Solutions of Singularly Perturbed Equations (in Russian) (Nauka Moscow 1973). GENERAL DISCUSSION 1 V Again J(v) is defined and there will be a value v* at which J(u*) = 0 and J changes sign. Let If the reacting mixture is confined to a finite region say 0 < x < 1 with boun- dary conditions imposed on u and v at both ends we proceed by solving with the given boundary conditions for v imposed upon V.This problem is easily analyzed by phase-plane arguments despite the discontinuity on the right. If there exists a solution V(x) crossing the value v* at some point x* then one can expect a solution (u,v) of the original problem when D is small for which u(x) is close to V(x) and for which u exhibits a transition in a small interval of width of the order D* at x = x*. On one side of the interval u is approximated by $,,( V(x)),and on the other side by t,h1(V(x)). The proof needs some extra minor assumptions. Again the solution appears to be stable if J > 0 in phase $o and J < 0 in The following example is chosen for clarity rather than reference to any physical system.The function f is not S-shaped but the above results hold in any event. The fact that negative concentrations are assumed can easily be remedied. Primes denote differentiation in x. Example Du” = (u-u)(u2-1) v’’ = -au u’ = v’ = Oatx = Oand 1. Here $,,(v) = -1 ql(u) = 1 u* = 0 and V(x)is piecewise parabolic. On the other hand if the medium is infinite (or large compared to the square root of v’s diffusion coefficient) we niay seek a solution passing from one stable GENERAL DISCUSSION stationary state (uo,uo) to another (ul q). It is reasonable to assume that u0 = t,b0(q,) u1 = @,(u,). Then the problem reduces to finding a solution V(x) of (I) exhibiting a phase transition from vo to ul.This is like the one dimensional problem in the paper presented. It may not have a solution unless there is a dependence on some parameters A,and d is chosen appropriately. 4. The procedure in (2) and (3) above can be carried out formally when there is more than one space dimension and the region is bounded. One sometimes obtains a cell of one phase surrounded by the other phase (nucleation). The mathematical details and proof for case (2) are given in ref. (3). 5. The nucleation problem discussed in the presented paper is such that when D is small the radius of the cell Rc is of the order D3 as is the transition layer width 1. The cells constructed as in (4) however have size independent of D for small D. Dr. J. S. Turner (Austirz Texas) said I would like to point out that nucleation by spontaneous localized fluctuations in systems exhibiting multiple steady states has been studied by Lefever Prigogine and myself.We have discussed the occurrence of metastable states in a simple model and have obtained a critical coherence length for spontaneous fluctuations necessary to form an unstable nucleus in an initially homogeneous metastable phase. For details see my discussion remark following the paper of Nicolis and Prigogine in this Symposium.