Self-replicating spots Int. J. Bif. Chaos. 16(12), (2007)

The major step  forward in the modern theory  of pattern formation was given by Turing (1952), who  used the linear analysis to determine the  conditions necessary  for  the creation  of  spatial patterns  in two-component reaction-diffusion  systems. A more  recent criteria for pattern formation  was proposed  by Koch & Meinhardt (1994) and Gierer &Meinhardt (2000) and  independently by Segel and Jackson (1972). They postulate that the interplay between two antagonic feedbacks is essential  for pattern formation.   On one hand,  the  positive  feedback  consists in  the  self-enhancement  or autocatalysis  of one of  the chemical components -  generally called activator  -,  a reaction  necessary  for  small  perturbations to  be amplified.    On  the   other  hand,   the  increase   in  activator's concentration  must be complemented by  a fast-diffusing  response in order to  obtain pattern  formation. 

The  most studied  examples  of the  two  types of  reaction-diffusion systems  are   the  Meinhardt  system  (Gierer &Meinhardt 2000)   and  the  diffusive Gray-Scott system (Pearson 1993) , respectively. The complex interplay between activator and inhibitor or substrate chemical, aided by  the  reaction  and  diffusion components  create  most  startling spatio-temporal  patterns, such as  spots, stripes ,  travelling waves , spot replication,  and spatio-temporal chaos,  in a nutshell,  a clear example of Turing patterns.   The Turing patterns are characterized by the active role that  diffusion plays in destabilizing the homogeneous steady state of the system. They emerge spontaneously as the system is driven  into  a state  where  it is  unstable  towards  the growth  of finite-wavelength stationary perturbations. Interesting  enough, the replication  characteristic  is   a  particularity  of  the  diffusive Gray-Scott  model   alone,  which  makes   it the  ideal   model  for developmental research.  In such cases, cell-like localized structures grow,  deform and  make replica  of themselves  until they  occupy the entire space .

The  Turing patterns  from the  work  of  Pearson 1993  on the  diffusive Gray-Scott  model   were  confirmed  experimentally   by  Lee, McCormick, Ouyang & Swinney (1993) including the spot  replication - Lee, McCormick, Swinney & Pearson (1994).  Theoretically, extensive work exist in the literature  on the dynamics of this model concerning the ``spot  replication'' in one, two and  three dimensions (Muratov & Osipov 2000). Moreover, Muratov & Osipov  (1999) developed a  theory of rotating spiral waves  for the diffusive Gray-Scott  model, as an  example of a vivid phenomenon  observed in many models and  biological systems.  In addition,  Nishiura &Ueyama (1999)  proposed a  theoretical mechanism that drives the replication  dynamics itself from a global bifurcation point of view.

This model was originally introduced in Gray-Scott (1985) as an isothermal system with chemical feedback  in a  continuously fed, well-stirred  tank reactor, where the last property implied the lack of diffusion. The analysis of the system revealed stationary states, sustained oscillations and even chaotic behavior. The model considers the chemical reactions describing  the  autocatalytic  growth   of  an  activator  V  on  the continuously fed substrate, U and the decay of the former in the inert product P, subsequently removed  from the system.  A major development was performed by Pearson (1993)  who introduced the role of space by  relaxing the constraint  of a well-stirred tank  and studied the evolution of the concentrations   of  the  two  chemical  components, u(x,y,t) and v(x,y,t) in two dimensions, in  the limit of small diffusion. The system has as parameters the Du and  Dv,  the  diffusion coefficients,  F, the dimensionless flow rate (the inverse of the residence time) and k, the decay  constant of  the activator,  V. The  original  study involved fixed   diffusion  coefficients,   Du  =   2  x 10 -5  and Dv=10 -5, with F and k being the control parameters.

For an  overview of the  resultant patterns, one can integrate the equations superimposing on the 2D space (X,Y)  a  gradient  of  the control  parameters, k and F. In such a way, one has in each cell an approximate sample of the resultant pattern for the associated F and k parameters. The  resultant  ``phase diagram''  is  shown on the right, where besides spot replication and stripes, the system shows for the bottom pairs of (F,k)
travelling waves and spatio-temporal chaos.

As discussed also  in Pearson (1993), the  spots occur only for the  parameter values for which  the only steady state  is the red one -  r.h.s. of  the saddle-node bifurcation  curve --, and  thus the gradient needed for the formation and maintenance of the blue spots is an intrinsic  self-sustaining feature  of the system.  Once a  spot of high V is  formed, it is maintained by  the concentration difference between  the its  center  and the  surroundings  of the  spot. As  the concentrations are limited to the  [0,1] interval, the spot can grow until  its  maximal  gradient  is  not enough  to  achieve  a  maximum concentration  in  the  center   and  thus  its  V-value  starts  to decrease. This induces the spot-division phase.

Mazin et al.(1996) account for and illustrate the Turing space - the region in the  parameter space for which the blue state is unstable with respect to the growth of standing spatial perturbations. Their  analysis clearly determines  the crucial  role  played by  the ratio of the diffusivity coefficients in the pattern formation. The increase of the diffusivity  ratio  beyond 2 , the value investigated  in the  original paper of Pearson(1993), leads to a  significant extension of the Turing regime.

Lesmes et al. (2003) have carried out the first study  of the noise-controlled pattern formation  in the Pearson model, with emphasis on the self-replicating patterns. They found that for the chosen value of the  pair (F,k), the noise drives the system from the non-multiplicative, stripe-like pattern to the spot-multiplication one .  Interesting enough, they argue in favor of an optimal noise intensity, A for which the number of spots  is maximal, after  a sufficiently long integration  time .

In comparison with the results of Lesmes et  al.,  ours  suggest  the  existence  of  an  interval  of  optimum noise-intensity values  leading to a  maximum number of  spots, rather than  a single  optimum value.   In other  words, our  distribution of spots shows a  plateau at the level of maximum  number of spots, while theirs  appears   to  have  a   more  bell-like  shape.   The  plateau  characteristics  is  more evident  in  the  distribution  of the  area occupied  by the  two  features --  spots  and lines  -  as the  noise intensity varies. 

Recently, Robert Munafo drew my attention to additional interesting studies of this system.

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