D. Theory of Excitable Media Excitability and Spatial Order in Membranes of Developing Systems BY B. C. GOODWIN School of Biological Sciences University of Sussex Brighton Sussex BNl 9QG Received 15th July 1974 A simple biological model of metabolic activities on membranes is explored with respect to excitability and spatial gradient formation. The model is shown to have the property of undergoing a phase transition from a non-excitable to an excitable condition. Once excitable it is shown that a spatially-stable gradient can be formed on the membrane. Periodic gradients can be formed by a slightly modified process. 1. INTRODUCTION The primary concern of this paper is the exploration of some temporal and spatial consequences of certain very elementary properties of biological membranes.The term " elementary " is employed in a biological not a physico-chemical sense. Thus it is taken as axiomatic that biological membranes are complex lipo-protein structures and that the proteins are as important as the lipids. This is the emphasis that has emerged in recent years as a result of the molecular biological approach to membrane structure and activity.l It is therefore assumed from the outset that membranes contain enzymes and proteins capable of reacting with specific ligands. Then on the basis of some very simple postulates about interactions between these proteins a model of metabolic excitability as a phase transition is presented ;ultimately it is shown how the postulates give rise to processes which result in either monotonic or periodic gradients of "morphogens " on the membrane.The reason for concentrating attention on biological membranes as the sites of spatial order in developing organisms is provided by experimental evidence which points very strongly in this direction. 2-6 Furthermore unicellular organisms show essentially the same morphogenetic capacities and behaviour as do the multicellulars so that a basic understanding of morphogenetic mechanisms must be arrived at independently of assumptions about cellular partitions. These points have been argued out more fully in a paper by Goodwin and McLaren in which the basic features of the model considered in this paper have been presented together with applications to particular developmental processes.The present goal is to look at membrane excitability from a particular perspective and to explore some further consequences of the basic model thus extending its range of applicability. 2. EXCITABILITY AS A PHASE TRANSITION The type of phenomenon now considered is the rather sudden appearance of spontaneous action potentials during the regeneration of the unicellular marine alga Acetabularia as reported by Novak and Bentrup.* Other developing systems show 226 B. C. GOODWIN similar types of state transition also but the electrophysiological behaviour of AcetabuZaria is particularly clearly defined. Whether this change of membrane state is correctly described as a phase transition of the type here described requires closer experimental study but the model draws attention to this possibility and hence to the eventuality that such transitions are an important aspect of development.OUTSIDE INSIDE FIG.].-Model of interactions between enzymes El and E2 metabolites U V and W,and the ionic species K+ which can result in propagating metabolic activity waves and the formation of a morphogenetic gradient on a membrane. For explanation see text. It is convenient first to describe the molecular organization within the membrane which characterizes the metabolically excitable condition underlying the model. This is shown in fig. 1. The circles labelled El lying within the membrane represent enzymes whose active sites face the interior of the cell. They catalyze the conversion of a metabolite U into another metabolite V.A second enzyme EZ,which is either a soluble enzyme or is loosely associated with the membrane converts V into W. It is assumed that an ionic species represented in the figure by K+,is in equilibrium between a membrane bound state and a free state and that V has the effect of displacing this equilibrium towzrds the free state. The ion is an activator of El so that a positive feed-back loop is created which introduces an instability into the kinetics. W on the other hand is an inhibitor of El so that the control circuit can be stabilized. A kinetic system of this type belongs to the same category of reaction system as the glycolytic oscillator extensively studied by Hess and his colleague^,^-^^ and to the general class of kinetic processes to be considered by Franck l1 in this symposium.Since there are many possible kinetic realizations of such a system there is no need to make particular assumptions ; one simply supposes that propagating waves of metabolic activity can occur providing certain constraints are satisfied. Several contributions to this Symposium are concerned with the definition of these constraints. It is evident that one condition which must be satisfied in order for there to be wave propagation in such a system is that the density of enzymes El within the membrane must be sufficiently large. The exact value will depend upon the stoichio- metries of the reactions the diffusion constants of the reacting species and of the activating ion the temperature etc.The density or the concentration of El in the membrane will depend upon the balance between the rate of its incorporation into the membrane and its rate of loss or dilution (if new membrane is being formed). All these factors will be represented by means of a biochemical potential p so that the concentration of enzyme is determined by this quantity. We may assume that El is effectively inactive until it takes the spatial configuration resulting from incorporation into the membrane so that the cytoplasm is not excitable. It is assumed that there are sites for enzyme El on the membrane and that their mean number per unit area is constant. The binary variable El is used to designate the state of site i on the membrane. This variable takes the value 0 or 1 accord-ing as the site is unoccupied or occupied respectively.The model now follows S 9-8* EXCITABILITY AND SPATIAL ORDER closely the lattice gas description of the process of condensation which is in turn based upon the Ising model of ferromagnetism using the mean field approximation.12* l3 This assumes that each unit in the system under consideration (in our case an enzyme) experiences a mean field of force due to the presence of other units and the essence of the procedure is to find an equation which expresses this field in terms of itself (the self-consistent field equation). The field under consideration in this model is assumed to arise from interactions between enzymes occupying membrane sites and those enter- ing the membrane the presence of an enzyme at a site facilitating the entry of an enzyme into the membrane at a neighbouring site.We designate this field as p = (Q the mean value of the occupancy averaged over all the sites. Let the total number of occupied sites be N so that Cci = N the sum being over all sites. And finally let the interaction between two neighbours be tii,. The appropriate function to use in determining average values in such a system is the grand partition function which allows for variations in both N the total number of active enzymes on the membrane and in the energy of the system resulting froin interactions. The Hamiltonian for such a system is then H = -+CU,~E~E~ -Np. (2.1) The self-consistent equation for p which is obtained from this expression via the grand partition function is (2P -I> = tanW?/2(pu(O)+PI (2.2) where u(0)is the Fourier transform of Cuijtaken at the point 0.The assumption here is that the interaction field is everywhere the same uij(R)being translation invariant. The parameter /? is l/kT. Thus it is assumed that the system is at a defined temper- ature. This temperature in fact implies that the relevant processes inolved are quasi-equilibrium ones in accordance with the use of a biochemical potential to represent the factors which determine the concentration of enzyme E in the mem- brane. Other processes such as those involved in the maintenance of the membrane and of a constant pool of precursor U are taken as constant parameters of the system. The consequences of this model are as follows.There is a region of parameter space in which eqn (2.2) has a single root and another region where it has three. The latter condition which is the interesting one for phase transitions requires that p = -u(0)/2 whence eqn (2.2) becomes (2p -1) = tanh[&(O)(2p -l)]. (‘2.3) For sufficiently large /? this equation has three roots in p one of which is always p = 3 and the other two are symmetric about this. The negative value of the potential in this region has no physical significance since this quantity is defined relative to an arbitrary constant. Plotting p as a function of T one gets a curve such as that shown in fig. 2. The region of triple roots lies below T, the critical temperature and the curve shown in this region is called the coexistence curve.Above T, no excitability is possible but below it a transition from the non-excitable to the excitable state can occur. Thus if the membrane is in the state shown by P and p is then increased (e.g. more enzyme is incorporated into the membrane) then the system can move to P’ an unstable point on the coexistence curve. A sudden state transition to Q’ can then occur so that the membrane becomes excitable. As the temperature is decreased this transition can occur at smaller values of p; i.e. thermal noise interferes less and less with the processes resulting in excitation. Now clearly in a biological system there will also be a lower temperature bound to the domain of excitability as well. Thus the coexistence curve should close on itself B.C. GOODWIN again. This could be modelled by making further assumptions about say the temperature-dependence of the quantities u(0) and p at lower values of T. However it is far from clear as yet that we are dealing with true phase transitions in excitability phenomena so a further analysis does not seem worthwhile. The point of this model is to demonstrate the possibility of such behaviour in membranes. The analysis could also be extended to a consideration of metastable regions of the coexistence domain corresponding to super-cooled and super-heated states. These would be conditions where very small fluctuations cause either excitation or the loss of excitation respect- ively. The former might describe the state of membranes which show latent excit- ability responding to small stimuli but are not recurrently active as in a pacemaker system.I I I T TC FIG.2.-Coexistence curve for non-excitable (1) and excitable (2) states on a membrane with molecu- lar interactions of the type shown in fig. 1. The region above Tccorresponds to a condition in which no transition to excitability can occur but below Tc such transitions are possible. The dotted line from P’ to Q’ describes such a transition which involves a rapid change in the variable p the fraction of the total number of sites which is occupied. 3. EXCITABLE MEMBRANES AND SPATIAL GRADIENTS Attention is now directed to certain spatial consequences of the model described in fig. 1. It is of interest to ask under what conditions activity waves can leave “ mem-ory ” traces on membranes in the form of gradients of a metabolite.Such a process would clearly be of significance in relation to both morphogenetic and neural activities. A simple mechanism for achieving this is to assume the existence in the membrane of proteins with binding sites for the metabolite V. These are shown as circIes marked B in fig. I. In order to get a graded distribution of bound V hereinafter referred to as morphogen we may assume that these sites are activated after the production of V by E Thus for example we could suppose that W is an activator of B as well as an inhibitor of El. This temporal asymmetry of the processes which constitute the propagating wave of activity on the membrane results in a graded distribution of rnorphogen with its maximum at the point of origin of the wave.’ To illustrate the principles of the model consider the following local representation.A pulse of metabolite of magnitude M is produced at the point x = p on a one-dimensional membrane as shown in fig. 3. The wave of activation of the binding sites is repre- sented by a pick-up function which starts at x = p and propagates with velocity v in the positive direction. We may then ask what the distribution of morphogen will be EXCITABILITY AND SPATIAL ORDER after the passage of the wave. The correct representation of the process involves the propagation of the pulse M as well as is done in another publication,’ but its essence is illustrated by the local model.X P FIG.3.-Local representation of the gradient-generating process on a membrane. The diffusible metabolite is produced in a pulse of magnitude Mat the point x = p on a one-dimensional membrane and the pick-up function (f)propagates from the point x = p at velocity D in the positive direction. The result is a gradient of bound metabolite represented by the function Y(x,a). The behaviour of the pulse of metabolite as a function of distance from the origin x and time t is given by the function M X(x t) = -== exp( -(p -~)~/4Dt) J4.nDt where D is the diffusion constant. The pick-up function is represented by f(s) = Kas exp( -as) s 2 0 =o s<o where s is the distance from the wave-front 01 is a parameter determining the distance from the wave-front to the maximum off($) which is Ke-l at s = l/a and K is the product of the concentration of binding sites and a rate constant for bound complex formation.The expression for the morphogen concentration Y(x,t),assuming that the bound complex is stable and that Y(x,0) = 0 is where the lower limit ro = 0 for x < p = x-p for -Y > p. Defining p(p-x>/z; and 7 = (p-x)~/~D, this becomes Y(x t) = C dr(z 4-fl)r-i cxp( -m(z +/I)-yir) 1:. where MavK c=--J4YD. For the case ro = 0 letting t+co this integral is j4 (3.4) B. C. GOODWIN 23 1 where 5 = p-x 2 0. For x > p the integral does not permit a simple closed form but it is easily shown that Y(x,co)decreases monotonically. The peak of morphogen occurs at a distance to= cr/cr(a+a) (where CT = a+ Jau/D) to the left of the origin of the pulse as shown in fig.3. Biologically plausible values of the parameters are v = 8 pm s-l o! = 0.025 pm-1 and D = 200 s-l giving j = 5.3 pm. Now if the temporal asymmetry is reversed so that the activation of the binding sites precedes the activation of enzyme and production of V then instead of (3.3) one obtains for Y the expression dx(Lir+p-x)t-* exp -a(vz+p-x)--402 * This represents a situation in which the pulse of metabolite occurs at the origin x = 0 while the pick-up function starts as before at x = p. Using the same procedure as previously one finds for the case T~ = 0 (x < p) This function increases monotonically up to p so that the maximum occurs for a value of x > p.Thus we see that the temporal relationships between metabolite production and pick-up determine which way the gradient of morphogen is formed. 4. PERIODIC DiSTRIBUTION OF MORPHOGEN In order to have spatially stable distributions of morphogen on membranes it is necessary to have well-defined self-stabilizing origins of wave initiation. This requires a further postulate in the model but this is easily provided by a consideration of how the morphogen is likely to act. It is reasonable to suppose that directly or indirectly it affects membrane properties as discussed more fully elsewhere. One such effect could be on the concentration of free K+itself. If the morphogen increases this concentration then the wave will tend to recur at the wave origin where the morphogen concentration is maximal.The kinetics of the system described in fig. 1 can then result in regularly recurring activity waves which will maintain dynamically a spatial gradient of morphogen. A spatially periodic distribution of morphogeii can result from the recurrent propagation of metabolic activation waves and binding-site activation waves if these propagate at different velocities from a common origin. For the waves will then pass one another at defined points on the membrane where their temporal relationships will be reversed. Thus the gradient will form in opposite directions on either side of these passing points. However in order for the waves to propagate independently of one another at different velocities it is necessary that the processes involved in metabolits production be different from those involved in binding site activation.Therefore the picture of fig. I must be modified so that each process has kinetics of the general type described for metabolite production. This could be achieved by assum- ing another pair of enzymes coupled as are El and E? with an ion activator distinct from K+. Then both these processes could be initiated by cz spatially-localized pacemaker region established by a gradient formed by a previous wave process. This complication of membrane-localized activities in no way overburdens this organelle with specific structure since a mitochondria1 membrane for example is very much more complex than this. The biological attraction of the above inodel is that it provides a plausible mech-anism for both aperiodic and periodic gradient formation in either unicellular or EXCITABILITY AND SPATIAL ORDER multi-cellular organisms.Furthermore having established a wave origin the spatially periodic process is a reliable one unlike the difficulties encountered by the Turing model in this respect. However although aperiodic gradients generated by the model regulate (are size-independent),' the periodic gradients do not ; so here the model fails biologically as does also the Turing process. This paper was written while the author held a Leverhuline Visiting Professorship at the Department of Theoretical Physics National University of Mexico and grateful acknowledgement is made to the Royal Society and to Professor M.Moshinsky Head of the Department for assistance and hospitality. C. Gitler Ann. Rev. Biophys. Bioeng. 1972 1 51. A. S. G. Curtis The Cell Surface (Academic Press London 1967). H. F. Stumpf Dev. Biol. 1967 16 144. D. L Nanney Science 1968 160,496. L. S. Sandakhchiev L. I. Puchkova and A. V. Pikalov Biology and Rodiobiofogy of Ai~nclcate Systems ed. S. Bonnotto 1972 p. 297. W. Herth and K. Sander Wilhelnt ROUX'Arch. Ent. Org. 1973 172 1. 'B. C. Goodwin and ID. I McLaren J. Theor. Biof. 1975. B. Novak and F. W. Bentrup Pfanta (Berl) 1972 108 227. B. Hess H. Kleinhaus and D. Kuschmitz Biological and Biochemical Oscillators ed. B. Chance and K. Pye (Academic Press New York 1973) p. 253. io A. Boiteux and B. Hess paper at this Symposium. * U. F. Franck paper at this Symposium. l2 R. Brout Phase Transitions (Benjamin New York 1965) Chap. 2 and 3. H. E. Stanley Introduction to Phase Transifions and Critical Phenomena (Clarendon Oxford 1971) chap. 6 and appendix A. I4 I. S. Gradsteyn and I. M. Ryzhik Tables of Integrals Series and Products (Academic Press New York 1965) p. 340. B. C. Goodwin Analytical Physiologj~ of Cells arid Detlefoping Organisins (Acadeiiiic Press 1976) chap. 5.