a chemical approach to designing turing patterns in reaction-diffusion systems

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  • Proc. Nati. Acad. Sci. USAVol. 89, pp. 3977-3979, May 1992Chemistry

    A chemical approach to designing Turing patterns inreaction-diffusion systems

    (pattern formation/nonlinear dynamics)

    ISTVAN LENGYEL*t AND IRVING R. EPSTEIN*:*Department of Chemistry and tCenter for Complex Systems, Brandeis University, Waltham, MA 02254-9110; and Institute of Physical Chemistry, KossuthLajos University, H-4010, Debrecen, Hungary

    Communicated by Richard M. Noyes, January 24, 1992 (received for review October 28, 1991)

    ABSTRACT A systematic approach is suggested to designchemical systems capable of displaying stationary, symmetry-breaking reaction diffusion patterns (Turing structures). Thetechnique utilizes the fact that reversible complexation of anactivator species to form an unreactive, immobile complexreduces the effective diffusion constant ofthe activator, therebyfacilitating the development of Turing patterns. The chlorinedioxide/iodine/malonic acid reaction is examined as an ex-ample, and it is suggested that a similar phenomenon mayoccur in some biological pattern formation processes.

    The stable, stationary, reaction-diffusion patterns predictedby Turing (1) have served as the basis for several theories ofmorphogenesis (2, 3). They have recently been found exper-imentally in the chlorite/iodide/malonic acid (CIMA) systemin a gel reactor (4). The difficulty in observing Turing patternsin other systems arises from the necessary condition that thediffusion coefficients of the activator and inhibitor species bevery different. Here, we show that introduction of a reagentthat reversibly forms an unreactive, immobile complex withthe activator species can circumvent this requirement, mak-ing it possible to produce Turing patterns in systems thatwould not otherwise exhibit such structures.The experimental difficulty of demonstrating the chemical

    nature of the postulated morphogens has prevented directstudy of the Turing mechanism in biological systems. Chem-ical reactions, in which the relevant species can be identifiedand monitored, offer far better prospects for investigatingTuring structures. The need, however, for the activator-i.e., the species responsible for the positive feedback-todiffuse significantly less rapidly than the inhibitor (3) hasposed a major obstacle, since in aqueous media nearly allsimple molecules and ions have diffusion coefficients withina factor of 2 of 1.5 x 10-5 cm2.s-.

    A General Two-Variable Model

    In the CIMA reaction, we have suggested that interactions ofthe activator iodine species with the polyacrylamide geland/or with the starch indicator result in a major reduction ofthe effective activator diffusion coefficient (6). Here wegeneralize that idea to an algorithm for designing systems thatexhibit Turing instability. Consider a system characterizedby parameters (e.g., rate constants) p and containing anactivator species X and an inhibitor species Y that react anddiffuse according to Eq. 1.

    ax a2x-=f(x, y, p) + Dx -at az2

    ay a2y-= g(x, y, p) + Dy 2at az

    af af agall --- L2- ,a2y - ax

    I1]ag

    a22 -aay

    where D. and Dy are the respective diffusion constants,lowercase letters represent concentrations, and the elementsaij of the Jacobian matrix are evaluated at the steady state ofinterest. The identification of X and Y as activator andinhibitor, respectively, implies that all > 0 and a22 < 0Suppose now that X can react to form a complex with

    another species S, which is either bound to a gel matrix or isso large that the diffusivities of S and the complex SX areeffectively zero. Suppose also that the complex is unreac-tive, either because of its chemical properties or becausesteric factors prevent Y from approaching X when it is boundto the large molecule. For simplicity we assume a largeexcess of S (s >> x) uniformly distributed so that itsconcentration is always very close to its initial value so. Thecomplex formation is described by

    sx k+X+SSX,' K= k-' K'=Kso.

    SIX k-I [2]

    If the spatial distribution of S is uniform, the new reaction-diffusion system is described by

    ax a8x-=f(x, y, p)-k+sox + kLsx + Dx -z2atXaay a2yat- (x y, p) +Dy az2

    asx- = k+sox -ksx.

    [3a]

    [3b]

    [3c]

    Ifthe formation and dissociation ofthe complex are rapid, theconcentration ofSX instantaneously follows that ofX, and sxcan be expressed in terms of x by using the last two expres-sions in Eq. 2.11 The set of differential equations can then besimplified by adding Eqs. 3a and 3c and replacing sx by K'x.

    Abbreviation: CIMA, chlorite/iodide/malonic acid.This is paper no. 76 in the series "Systematic Design of ChemicalOscillations." Paper no. 75 is ref. 5.lHunding and Sorensen (7) have suggested a similar approach todescribe size adaptation of Turing patterns in developing embryos."This reduction can be made rigorous using singular perturbationtheory by letting k+ = k'/E, kL = kL/E, where k+ and kL are O(1),and then letting E -- 0.

    3977

    The publication costs of this article were defrayed in part by page chargepayment. This article must therefore be hereby marked "advertisement"in accordance with 18 U.S.C. 1734 solely to indicate this fact.

  • 3978 Chemistry: Lengyel and Epstein

    a(x + sx) ax a2x= (1 + K') - =f(x, y,p) + Dx-

    at at aZ2

    ay a2y= g(x,y,p) + Dyat az2

    a22

    [4]

    This two-variable reaction-diffusion system is more easilyanalyzed if one rescales the time and length variables bysetting t = t'/(1 + K'), z = D,12z'. The differential Eq. 4 thentakes the form

    ax adx- =f(x, Y, P) + -e2at' az'2ay = ay

    oi-0,g(x, y, p) +c-Ia2[5]

    where oa = 1 + K' and c = Dy/DX. Observe thatf, g, and pretain the same form. Whether or not the original differentialequation system (Eq. 4) is in dimensionless form, the com-plex formation (Eq. 2) and the subsequent transformationchange neither the scaling of the dependent (concentration)variables nor the kinetic parameters of the system.** Phys-ically, formation of the complex separates the time scales forchanges in the activator and inhibitor concentrations by afactor of o. In particular, the more stable the complex, thegreater the difference between the time scales for the diffu-sion of the two components.Complex formation does not change the steady-state com-

    position of the system. It may, however, modify the stabilityof the steady state. If the system has a unique unstable steadystate and hence shows oscillatory behavior in the absence ofS (o' = 1), then complex formation (a > 1) can stabilize thehomogeneous steady state at this same set of parameters.Introduction of S thus broadens the range of steady-statestability as illustrated in Fig. 1, making it possible fordiffusion-induced instability to occur at parameter values thatwere inside the oscillatory region ofthe complex-free system.Equivalently, complex formation may be viewed as shiftingthe homogeneous Hopf bifurcation to larger parameter val-ues, so that the Turing bifurcation occurs before the Hopfpoint and is therefore observable. The wavelength of theTuring patterns and the stability ofthe inhomogeneous steadystate are independent of the concentration of complexingagent, a result in agreement with observations (P. De Kepper,personal communication) on the CIMA/starch system withvarying concentrations of starch.

    The CIMA System

    We illustrate these ideas with reference to the CIMA reaction(4). More general and detailed treatments of the Turingbifurcation can be found in refs. 3 and 8. The system is welldescribed by three overall stoichiometric processes whoserate laws have been experimentally determined. After somesimplifications, the dynamics can be summarized in a two-variable model (9):

    A-*X v1=kl

    X- Y 2=k2[X]

    [X][Y]X+Y-*P v3=k~u+ [X]2

    k' = kl[MA]o

    k'= k2[C102] [6]

    k' = k3[12]0 ,

    ~~~~,.all

    ~~~a110'> 1

    o'= 1

    FIG. 1. Stability boundary for a general activator-inhibitor modelin the all vs. a22 space. The hatched area indicates the stable steadystate.

    where X and Y denote iodide and chlorite ions, the activatorand inhibitor, respectively, and MA denotes malonic acid.The starch reacts with iodine and iodide to form a blue

    helical complex. Because of its high molecular weight, thestarch is essentially immobile either in the gel or in aqueoussolution. In a water solution, formation of the complex is avery fast process, though it may be somewhat slower in amedium like the polyacrylamide gel.We assume rapid formation of a starch/triiodide complex

    SI- according to

    S + I- + 12 = SI3[SI]

    K=IS][12][I-]' [7]

    where [S] represents the concentration of starch per sixmonomer units needed to form a complete helix (10, 11). Thedimensionless rate Eq. 4 for the system allowing for complexformation (Eq. 7) takes the form (cf. Eq. 5):

    ax xy a2x-=a-x-4 +aT 1 +x2 az2

    [8]ay xy ay2-=oi b x - 2 +caT V 1+x2 az2.

    Fig. 2 shows, for realistic values of the rate parameters andconcentrations, the range of diffusion-induced instability atvarious values of a-. In the absence of starch, Cmin is 10,while with sufficient starch to give or = 8, roughly theconditions employed by Castets et al. (4), Cmin is reduced to1.5, a plausible value for the ratio oftwo small inorganic ions.Even in the absence of starch, the polyacrylamide gel may actas a complexing agent with similar effect.

    A Design Algorithm

    The above analysis suggests a path toward the elusive goal ofdesigning new systems that exhibit Turing patterns, a paththat may already have been followed by many biologicalsystems. By complexing an activator molecule either with amacromolecule or with a membrane-bound species, it be-comes possible to obtain Turing structures even when theratio c of diffusivities of inhibitor and free activator does not

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