Physics Letters A 371 (2007) 41–47 www.elsevier.com/locate/pla Cross-diffusion in the Templator model of chemical self-replication Jessica M. Chung, Enrique Peacock-López ∗ Department of Chemistry, Williams College, Williamstown, MA 01267, USA Received 14 February 2007; received in revised form 20 April 2007; accepted 26 April 2007 Available online 12 June 2007 Communicated by C.R. Doering Abstract For the Templator model of chemical self-replication, we consider the effect of cross-diffusion in pattern formation. Although other chemical systems with cross-diffusion have been reported, we are the first to obtain exact relations between the relevant parameters for the Poincaré– Adronov–Hopf and Turing bifurcations. Moreover, for the simplest case, we obtain analytical expressions in parameter space for the bifurcation curves and the Turing–PAH codimension two point.  2007 Elsevier B.V. All rights reserved. PACS: 87.10.+e; 05.70.Ln; 47.54.+r 1. Introduction Chemical self-replication is the means by which an individ- ual molecule can duplicate itself. A product molecule directs the assembly of two or more component molecules to form a new product molecule by acting as a template to position the components for the joining reaction [1,2]. There are two crit- ical steps in this process. First, product molecules must bind available components to facilitate the synthesis of more prod- uct molecules. Second, once the first reaction is complete, the product–product complex must readily dissociate so that the newly formed product molecule may join the other template molecules. In an ideal self-replicating system, as soon as the new template molecule is formed, it should readily dissoci- ate from the other template molecule and begin to act as a template by binding other component molecules. Since tem- plate molecules can participate in multiple replication cycles, the growth rate of the concentration of the template product molecule is directly proportional to the replicator concentration. This relationship yields an exponential growth rate for the con- centration of the template molecule; for sustained exponential growth this process is known as autocatalytic self-replication. * Corresponding author. E-mail address: epeacock@williams.edu (E. Peacock-López). In reality, however, it is difficult to achieve the delicate balance between strong binding of component molecules that facilitate the formation of a new product molecule and easy dissociation of product–product complex that is required for autocatalytic self-replication. The most significant complication is due to the difficulty of dissociation of the original template molecule from the newly formed template molecule. When template molecules remain together, the number of these molecules available to as- sist in the production of more template molecules is reduced and thus the growth rate of the product may be less than expo- nential [3,4]. The concept that species (prey) may diffuse due to the pres- ence of other species (predator) has been considered in eco- logical literature [5–8]. After all, you may try to avoid an in- fected individual during an epidemic. Therefore, a diffusion current due to gradients of other species seems more natural in macro-biological systems [9,10]. Although less intuitive, cross- diffusion is an observable and measurable property of solutions. Consequently, in our analysis of chemical self-replication we include now, the off-diagonal elements of the diffusion tensor or the so-called cross-diffusion [11–13] which has been measured and studied experimentally in multicomponent systems [14,15]. In particular, studies in ternary solutions enforce the following thermodynamic stability conditions for solution stability: (1a) D 1 + D 2 > 0, 0375-9601/$ – see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.04.114